diff --git a/src/core/include/mp-units/bits/reference_concepts.h b/src/core/include/mp-units/bits/reference_concepts.h
index c51d2d593..d3fdbaf35 100644
--- a/src/core/include/mp-units/bits/reference_concepts.h
+++ b/src/core/include/mp-units/bits/reference_concepts.h
@@ -77,6 +77,7 @@ concept ReferenceOf =
    (QuantitySpec<std::remove_const_t<decltype(V)>> && implicitly_convertible(get_quantity_spec(T{}), V) &&
     !detail::DerivedFromQuantityKindSpecOf<get_quantity_spec(T{}), V> &&
     (detail::QuantityKindSpec<std::remove_const_t<decltype(get_quantity_spec(T{}))>> ||
-     !detail::DerivedFromQuantityKindSpecOf<V, get_quantity_spec(T{})>)));
+     !detail::DerivedFromQuantityKindSpecOf<V, get_quantity_spec(T{})>)) ||
+   (std::same_as<std::remove_const_t<decltype(V)>, quantity_character> && (get_quantity_spec(T{}).character == V)));
 
 }  // namespace mp_units
diff --git a/src/core/include/mp-units/bits/representation_concepts.h b/src/core/include/mp-units/bits/representation_concepts.h
index f25a0d9b6..36328a964 100644
--- a/src/core/include/mp-units/bits/representation_concepts.h
+++ b/src/core/include/mp-units/bits/representation_concepts.h
@@ -52,35 +52,88 @@ enum class quantity_character { scalar, vector, tensor };
 
 namespace detail {
 
-template<typename T, typename U>
-concept CommonTypeWith =
-  std::same_as<std::common_type_t<T, U>, std::common_type_t<U, T>> &&
-  std::constructible_from<std::common_type_t<T, U>, T> && std::constructible_from<std::common_type_t<T, U>, U>;
+template<typename T>
+concept Scalar = is_scalar<T>;
+
+template<typename T>
+concept Vector = is_vector<T>;
+
+template<typename T>
+concept Tensor = is_tensor<T>;
 
-template<typename T, typename U = T>
-concept ScalableNumber =
-  std::regular_invocable<std::multiplies<>, T, U> && std::regular_invocable<std::divides<>, T, U>;
+template<typename T, quantity_character Ch>
+concept IsOfCharacter =
+  (Ch == quantity_character::scalar && is_scalar<T>) || (Ch == quantity_character::vector && is_vector<T>) ||
+  (Ch == quantity_character::tensor && is_tensor<T>);
+
+template<typename T>
+[[nodiscard]] consteval quantity_character get_character(T)
+{
+  if constexpr (Tensor<T>)
+    return quantity_character::tensor;
+  else if constexpr (Vector<T>)
+    return quantity_character::vector;
+  else if constexpr (Scalar<T>)
+    return quantity_character::scalar;
+}
+
+// clang-format off
+template<typename T>
+concept ScalarRepresentation = std::regular<T> && Scalar<T> &&
+  requires(T a, T b, std::intmax_t i, long double f) {
+    { a * i } -> Scalar;
+    { i * a } -> Scalar;
+    { a / i } -> Scalar;
+    { a * f } -> Scalar;  // TODO How this affects freestanding?
+    { f * a } -> Scalar;
+    { a / f } -> Scalar;
+    { a + b } -> Scalar;
+    { a - b } -> Scalar;
+    { a * b } -> Scalar;
+    { a / b } -> Scalar;
+  };
 
 template<typename T>
-concept CastableNumber = CommonTypeWith<T, std::intmax_t> && ScalableNumber<std::common_type_t<T, std::intmax_t>>;
+concept VectorRepresentation = std::regular<T> && Vector<T> &&
+  requires(T a, T b, std::intmax_t i, long double f) {
+    { a * i } -> Vector;
+    { i * a } -> Vector;
+    { a / i } -> Vector;
+    { a * f } -> Vector;
+    { f * a } -> Vector;
+    { a / f } -> Vector;
+    { a + b } -> Vector;
+    { a - b } -> Vector;
+    { dot_product(a, b) } -> Scalar;
+    { cross_product(a, b) } -> Vector;
+    { tensor_product(a, b) } -> Tensor;
+    { norm(a) } -> Scalar;
+  };
 
-// TODO Fix it according to sudo_cast implementation
 template<typename T>
-concept Scalable =
-  CastableNumber<T> ||
-  (requires { typename T::value_type; } && CastableNumber<typename T::value_type> &&
-   ScalableNumber<T, std::common_type_t<typename T::value_type, std::intmax_t>>) ||
-  (requires { typename T::element_type; } && CastableNumber<std::remove_reference_t<typename T::element_type>> &&
-   ScalableNumber<T, std::common_type_t<std::remove_reference_t<typename T::element_type>, std::intmax_t>>);
+concept TensorRepresentation = std::regular<T> && Tensor<T> &&
+  requires(T a, T b, std::intmax_t i, long double f) {
+    { a * i } -> Tensor;
+    { i * a } -> Tensor;
+    { a / i } -> Tensor;
+    { a * f } -> Tensor;
+    { f * a } -> Tensor;
+    { a / f } -> Tensor;
+    { a + b } -> Tensor;
+    { a - b } -> Tensor;
+    { tensor_product(a, b) } -> Tensor;
+    { inner_product(a, b) } -> Tensor;
+    { scalar_product(a, b) } -> Scalar;
+  };
+// clang-format on
 
 }  // namespace detail
 
 template<typename T>
-concept Representation = (is_scalar<T> || is_vector<T> || is_tensor<T>)&&std::regular<T> && detail::Scalable<T>;
+concept Representation =
+  detail::ScalarRepresentation<T> || detail::VectorRepresentation<T> || detail::TensorRepresentation<T>;
 
 template<typename T, quantity_character Ch>
-concept RepresentationOf = Representation<T> && ((Ch == quantity_character::scalar && is_scalar<T>) ||
-                                                 (Ch == quantity_character::vector && is_vector<T>) ||
-                                                 (Ch == quantity_character::tensor && is_tensor<T>));
+concept RepresentationOf = detail::IsOfCharacter<T, Ch> && Representation<T>;
 
 }  // namespace mp_units
diff --git a/src/core/include/mp-units/quantity.h b/src/core/include/mp-units/quantity.h
index 850cf624e..083da70fe 100644
--- a/src/core/include/mp-units/quantity.h
+++ b/src/core/include/mp-units/quantity.h
@@ -65,20 +65,6 @@ concept QuantityConvertibleTo =
   // deduced thus the function is evaluated here and may emit truncating conversion or other warnings)
   requires(QFrom q) { detail::sudo_cast<QTo>(q); };
 
-template<quantity_character Ch, typename Func, typename T, typename U>
-concept InvokeResultOf = std::regular_invocable<Func, T, U> && RepresentationOf<std::invoke_result_t<Func, T, U>, Ch>;
-
-template<typename Func, typename Q1, typename Q2>
-concept InvocableQuantities = Quantity<Q1> && Quantity<Q2> &&
-                              InvokeResultOf<common_quantity_spec(Q1::quantity_spec, Q2::quantity_spec).character, Func,
-                                             typename Q1::rep, typename Q2::rep> &&
-                              requires { common_reference(Q1::reference, Q2::reference); };
-
-template<typename Func, Quantity Q1, Quantity Q2>
-  requires detail::InvocableQuantities<Func, Q1, Q2>
-using common_quantity_for = quantity<common_reference(Q1::reference, Q2::reference),
-                                     std::invoke_result_t<Func, typename Q1::rep, typename Q2::rep>>;
-
 }  // namespace detail
 
 /**
@@ -341,24 +327,54 @@ explicit quantity(Q) -> quantity<quantity_like_traits<Q>::reference, typename qu
 
 // binary operators on quantities
 template<auto R1, typename Rep1, auto R2, typename Rep2>
-  requires detail::InvocableQuantities<std::plus<>, quantity<R1, Rep1>, quantity<R2, Rep2>>
+  requires requires(Rep1 a, Rep2 b) {
+    common_reference(R1, R2);
+    {
+      a + b
+    } -> detail::IsOfCharacter<get_quantity_spec(R1).character>
+  }
 [[nodiscard]] constexpr Quantity auto operator+(const quantity<R1, Rep1>& lhs, const quantity<R2, Rep2>& rhs)
 {
-  using ret = detail::common_quantity_for<std::plus<>, quantity<R1, Rep1>, quantity<R2, Rep2>>;
+  using ret = quantity<common_reference(R1, R2), decltype(lhs.number() + rhs.number())>;
   return make_quantity<ret::reference>(ret(lhs).number() + ret(rhs).number());
 }
 
 template<auto R1, typename Rep1, auto R2, typename Rep2>
-  requires detail::InvocableQuantities<std::minus<>, quantity<R1, Rep1>, quantity<R2, Rep2>>
+  requires requires(Rep1 a, Rep2 b) {
+    common_reference(R1, R2);
+    {
+      a - b
+    } -> detail::IsOfCharacter<get_quantity_spec(R1).character>
+  }
 [[nodiscard]] constexpr Quantity auto operator-(const quantity<R1, Rep1>& lhs, const quantity<R2, Rep2>& rhs)
 {
-  using ret = detail::common_quantity_for<std::minus<>, quantity<R1, Rep1>, quantity<R2, Rep2>>;
+  using ret = quantity<common_reference(R1, R2), decltype(lhs.number() - rhs.number())>;
   return make_quantity<ret::reference>(ret(lhs).number() - ret(rhs).number());
 }
 
+template<ReferenceOf<quantity_character::scalar> auto R1, typename Rep1,
+         ReferenceOf<quantity_character::scalar> auto R2, typename Rep2>
+  requires(!treat_as_floating_point<Rep1>) && (!treat_as_floating_point<Rep2>) && requires(Rep1 a, Rep2 b) {
+    common_reference(R1, R2);
+    {
+      a % b
+    } -> detail::IsOfCharacter<quantity_character::scalar>
+  }
+[[nodiscard]] constexpr Quantity auto operator%(const quantity<R1, Rep1>& lhs, const quantity<R2, Rep2>& rhs)
+{
+  gsl_ExpectsAudit(rhs.number() != quantity_values<Rep1>::zero());
+  using ret = quantity<common_reference(R1, R2), decltype(lhs.number() % rhs.number())>;
+  return make_quantity<ret::reference>(ret(lhs).number() % ret(rhs).number());
+}
+
 template<auto R1, typename Rep1, auto R2, typename Rep2>
-  requires detail::InvokeResultOf<(get_quantity_spec(R1) * get_quantity_spec(R2)).character, std::multiplies<>, Rep1,
-                                  Rep2>
+  requires(ReferenceOf<std::remove_const_t<decltype(R1)>, quantity_character::scalar> ||
+           ReferenceOf<std::remove_const_t<decltype(R2)>, quantity_character::scalar>) &&
+          requires(Rep1 a, Rep2 b) {
+            {
+              a* b
+            } -> detail::IsOfCharacter<(get_quantity_spec(R1) * get_quantity_spec(R2)).character>
+          }
 [[nodiscard]] constexpr Quantity auto operator*(const quantity<R1, Rep1>& lhs, const quantity<R2, Rep2>& rhs)
 {
   return make_quantity<R1 * R2>(lhs.number() * rhs.number());
@@ -366,21 +382,30 @@ template<auto R1, typename Rep1, auto R2, typename Rep2>
 
 template<auto R, typename Rep, typename Value>
   requires(!Quantity<Value>) &&
-          detail::InvokeResultOf<get_quantity_spec(R).character, std::multiplies<>, Rep, const Value&>
-[[nodiscard]] constexpr Quantity auto operator*(const quantity<R, Rep>& q, const Value& v)
+          (ReferenceOf<std::remove_const_t<decltype(R)>, quantity_character::scalar> || detail::Scalar<Value>) &&
+          requires(Rep a, Value b) {
+            {
+              a* b
+            } -> detail::IsOfCharacter<(get_quantity_spec(R) * detail::get_character(b)).character>
+          }
+
+detail::InvokeResultOf<get_quantity_spec(R).character, std::multiplies<>, Rep,
+                       const Value&> [[nodiscard]] constexpr Quantity auto
+operator*(const quantity<R, Rep>& q, const Value& v)
 {
   return make_quantity<R>(q.number() * v);
 }
 
 template<typename Value, auto R, typename Rep>
   requires(!Quantity<Value>) &&
+          (ReferenceOf<std::remove_const_t<decltype(R)>, quantity_character::scalar> || detail::Scalar<Value>) &&
           detail::InvokeResultOf<get_quantity_spec(R).character, std::multiplies<>, const Value&, Rep>
 [[nodiscard]] constexpr Quantity auto operator*(const Value& v, const quantity<R, Rep>& q)
 {
   return make_quantity<R>(v * q.number());
 }
 
-template<auto R1, typename Rep1, auto R2, typename Rep2>
+template<auto R1, typename Rep1, ReferenceOf<quantity_character::scalar> auto R2, typename Rep2>
   requires detail::InvokeResultOf<(get_quantity_spec(R1) / get_quantity_spec(R2)).character, std::divides<>, Rep1, Rep2>
 [[nodiscard]] constexpr Quantity auto operator/(const quantity<R1, Rep1>& lhs, const quantity<R2, Rep2>& rhs)
 {
@@ -405,14 +430,41 @@ template<typename Value, auto R, typename Rep>
   return make_quantity<::mp_units::one / R>(v / q.number());
 }
 
-template<auto R1, typename Rep1, auto R2, typename Rep2>
-  requires(!treat_as_floating_point<Rep1>) && (!treat_as_floating_point<Rep2>) &&
-          detail::InvocableQuantities<std::modulus<>, quantity<R1, Rep1>, quantity<R2, Rep2>>
-[[nodiscard]] constexpr Quantity auto operator%(const quantity<R1, Rep1>& lhs, const quantity<R2, Rep2>& rhs)
+template<ReferenceOf<quantity_character::vector> auto R1, typename Rep1,
+         ReferenceOf<quantity_character::vector> auto R2, typename Rep2>
+  requires requires(Rep1 v1, Rep2 v2) {
+    {
+      dot_product(v1, v2)
+    } -> detail::Vector;
+  }
+[[nodiscard]] QuantityOf<dot_product(get_quantity_spec(R1), get_quantity_spec(R2))> auto dot_product(
+  const quantity<R1, Rep1>& q1, const quantity<R2, Rep2>& q2)
 {
-  gsl_ExpectsAudit(rhs.number() != quantity_values<Rep1>::zero());
-  using ret = detail::common_quantity_for<std::modulus<>, quantity<R1, Rep1>, quantity<R2, Rep2>>;
-  return make_quantity<ret::reference>(ret(lhs).number() % ret(rhs).number());
+  return make_quantity<dot_product(R1, R2)>(dot_product(q1.number(), q2.number()));
+}
+
+template<ReferenceOf<quantity_character::vector> auto R1, typename Rep1,
+         ReferenceOf<quantity_character::vector> auto R2, typename Rep2>
+  requires requires(Rep1 v1, Rep2 v2) {
+    {
+      cross_product(v1, v2)
+    } -> detail::Scalar;
+  }
+[[nodiscard]] QuantityOf<cross_product(get_quantity_spec(R1), get_quantity_spec(R2))> auto cross_product(
+  const quantity<R1, Rep1>& q1, const quantity<R2, Rep2>& q2)
+{
+  return make_quantity<cross_product(R1, R2)>(cross_product(q1.number(), q2.number()));
+}
+
+template<ReferenceOf<quantity_character::vector> auto R, typename Rep>
+  requires requires(Rep v) {
+    {
+      norm(v)
+    } -> detail::Scalar;
+  }
+[[nodiscard]] QuantityOf<norm(get_quantity_spec(R))> auto norm(const quantity<R, Rep>& q)
+{
+  return make_quantity<norm(R)>(norm(q.number()));
 }
 
 template<auto R1, typename Rep1, auto R2, typename Rep2>
@@ -492,6 +544,7 @@ template<auto R, typename Rep, typename Value>
   return q <=> make_quantity<::mp_units::one>(v);
 }
 
+
 // make_quantity
 template<Reference auto R, typename Rep>
   requires quantity<R, std::remove_cvref_t<Rep>>::_rep_safe_constructible_
diff --git a/src/core/include/mp-units/quantity_spec.h b/src/core/include/mp-units/quantity_spec.h
index 0b3515081..ab6ce1325 100644
--- a/src/core/include/mp-units/quantity_spec.h
+++ b/src/core/include/mp-units/quantity_spec.h
@@ -275,19 +275,20 @@ struct quantity_spec<Self, Eq, Args...> : detail::quantity_spec_interface<Self>
  *       errors. Having them of the same names improves user experience and somehow blurs those separate domains.
  *
  * @tparam Q quantity specification of a parent quantity
- * @tparam Args optionally a value of a `quantity_character` in case the base quantity should not be scalar
- *              or `is_kind` in case the quantity starts a new hierarchy tree of a kind
+ * @tparam Args optionally `is_kind` in case the quantity starts a new hierarchy tree of a kind
  */
 #ifdef __cpp_explicit_this_parameter
-template<detail::NamedQuantitySpec auto QS, one_of<quantity_character, struct is_kind> auto... Args>
+template<detail::NamedQuantitySpec auto QS, one_of<struct is_kind> auto... Args>
   requires(... && !QuantitySpec<std::remove_const_t<decltype(Args)>>)
 struct quantity_spec<QS, Args...> : std::remove_const_t<decltype(QS)> {
 #else
+// template<typename Self, detail::NamedQuantitySpec auto QS, one_of<struct is_kind> auto... Args>
 template<typename Self, detail::NamedQuantitySpec auto QS, one_of<quantity_character, struct is_kind> auto... Args>
   requires(... && !QuantitySpec<std::remove_const_t<decltype(Args)>>)
 struct quantity_spec<Self, QS, Args...> : std::remove_const_t<decltype(QS)> {
 #endif
   static constexpr auto _parent_ = QS;
+  // static constexpr quantity_character character = QS.character;
   static constexpr quantity_character character = detail::quantity_character_init<Args...>(QS.character);
 
 #ifndef __cpp_explicit_this_parameter
@@ -488,6 +489,7 @@ template<QuantitySpec auto... From, QuantitySpec Q>
 // Operators
 
 [[nodiscard]] consteval QuantitySpec auto operator*(QuantitySpec auto lhs, QuantitySpec auto rhs)
+  requires(lhs.character == quantity_character::scalar || rhs.character == quantity_character::scalar)
 {
   return clone_kind_of<lhs, rhs>(
     detail::expr_multiply<derived_quantity_spec, struct dimensionless, detail::type_list_of_quantity_spec_less>(
@@ -496,6 +498,7 @@ template<QuantitySpec auto... From, QuantitySpec Q>
 
 template<QuantitySpec Lhs, QuantitySpec Rhs>
 [[nodiscard]] consteval QuantitySpec auto operator/(Lhs lhs, Rhs rhs)
+  requires(rhs.character == quantity_character::scalar)
 {
   return clone_kind_of<lhs, rhs>(
     detail::expr_divide<derived_quantity_spec, struct dimensionless, detail::type_list_of_quantity_spec_less>(
@@ -503,6 +506,7 @@ template<QuantitySpec Lhs, QuantitySpec Rhs>
 }
 
 [[nodiscard]] consteval QuantitySpec auto operator/(int value, QuantitySpec auto q)
+  requires(q.character == quantity_character::scalar)
 {
   gsl_Expects(value == 1);
   return clone_kind_of<q>(detail::expr_invert<derived_quantity_spec, struct dimensionless>(q));
@@ -516,6 +520,28 @@ template<QuantitySpec Lhs, QuantitySpec Rhs>
   return is_same_v<Lhs, Rhs>;
 }
 
+[[nodiscard]] consteval QuantitySpec auto dot_product(QuantitySpec auto lhs, QuantitySpec auto rhs)
+  requires(lhs.character == quantity_character::vector && rhs.character == quantity_character::vector)
+{
+  return clone_kind_of<lhs, rhs>(
+    detail::expr_multiply<derived_quantity_spec, struct dimensionless, detail::type_list_of_quantity_spec_less>(
+      remove_kind(lhs), remove_kind(rhs)));
+}
+
+[[nodiscard]] consteval QuantitySpec auto cross_product(QuantitySpec auto lhs, QuantitySpec auto rhs)
+  requires(lhs.character == quantity_character::vector && rhs.character == quantity_character::vector)
+{
+  return clone_kind_of<lhs, rhs>(
+    detail::expr_multiply<derived_quantity_spec, struct dimensionless, detail::type_list_of_quantity_spec_less>(
+      remove_kind(lhs), remove_kind(rhs)));
+}
+
+[[nodiscard]] consteval QuantitySpec auto norm(QuantitySpec auto q)
+  requires(q.character == quantity_character::vector)
+{
+  return q;
+}
+
 template<detail::QuantityKindSpec Lhs, detail::QuantityKindSpec Rhs>
 [[nodiscard]] consteval bool operator==(Lhs, Rhs)
 {
@@ -1475,7 +1501,8 @@ template<QuantitySpec Q>
 
 template<QuantitySpec Q1, QuantitySpec Q2>
 [[nodiscard]] consteval QuantitySpec auto common_quantity_spec(Q1 q1, Q2 q2)
-  requires(implicitly_convertible(get_kind(q1), get_kind(q2)) || implicitly_convertible(get_kind(q2), get_kind(q1)))
+  requires(Q1::character == Q2::character) &&
+          (implicitly_convertible(get_kind(q1), get_kind(q2)) || implicitly_convertible(get_kind(q2), get_kind(q1)))
 {
   using QQ1 = std::remove_const_t<decltype(remove_kind(q1))>;
   using QQ2 = std::remove_const_t<decltype(remove_kind(q2))>;
diff --git a/src/systems/isq/include/mp-units/systems/isq/electromagnetism.h b/src/systems/isq/include/mp-units/systems/isq/electromagnetism.h
index 981b0b051..c383d14fd 100644
--- a/src/systems/isq/include/mp-units/systems/isq/electromagnetism.h
+++ b/src/systems/isq/include/mp-units/systems/isq/electromagnetism.h
@@ -52,10 +52,10 @@ QUANTITY_SPEC(electric_flux_density, electric_polarization);  // vector
 inline constexpr auto electric_displacement = electric_flux_density;
 QUANTITY_SPEC(capacitance, electric_charge / voltage);
 // TODO how to calculate an argument of a vector product?
-QUANTITY_SPEC(magnetic_flux_density, force / (electric_charge * velocity), quantity_character::vector);
-QUANTITY_SPEC(magnetic_vector_potential,
-              magnetic_flux_density* length);  // vector // TODO what is a correct equation here?
-QUANTITY_SPEC(linked_flux, magnetic_vector_potential* displacement, quantity_character::scalar);
+// QUANTITY_SPEC(magnetic_flux_density, force / (electric_charge * velocity), quantity_character::vector);
+// QUANTITY_SPEC(magnetic_vector_potential,
+//               magnetic_flux_density* length);  // vector // TODO what is a correct equation here?
+// QUANTITY_SPEC(linked_flux, magnetic_vector_potential* displacement, quantity_character::scalar);
 QUANTITY_SPEC(magnetic_constant,
               electric_potential* time / (electric_current * length));  // TODO what is a correct equation here?
 inline constexpr auto permeability_of_vacuum = magnetic_constant;
@@ -64,7 +64,7 @@ QUANTITY_SPEC(speed_of_light, speed);
 inline constexpr auto light_speed = speed_of_light;
 QUANTITY_SPEC(electric_constant, 1 / (magnetic_constant * pow<2>(speed_of_light)));
 inline constexpr auto permittivity_of_vacuum = electric_constant;
-QUANTITY_SPEC(permittivity, electric_flux_density / electric_field_strength, quantity_character::scalar);
+QUANTITY_SPEC(permittivity, norm(electric_flux_density) / norm(electric_field_strength), quantity_character::scalar);
 QUANTITY_SPEC(relative_permittivity, dimensionless, permittivity / electric_constant);
 QUANTITY_SPEC(electric_susceptibility, dimensionless,
               electric_polarization / electric_constant / electric_field_strength, quantity_character::scalar);
@@ -73,13 +73,13 @@ QUANTITY_SPEC(displacement_current_density, electric_flux_density / time);  // v
 QUANTITY_SPEC(displacement_current, electric_current, displacement_current_density* area, quantity_character::scalar);
 QUANTITY_SPEC(total_current, electric_current);
 QUANTITY_SPEC(total_current_density, electric_current_density);  // vector
-QUANTITY_SPEC(magnetic_flux, magnetic_flux_density* area, quantity_character::scalar);
+// QUANTITY_SPEC(magnetic_flux, magnetic_flux_density* area, quantity_character::scalar);
 QUANTITY_SPEC(magnetic_moment, electric_current* area, quantity_character::vector);
 inline constexpr auto magnetic_area_moment = magnetic_moment;
 QUANTITY_SPEC(magnetization, magnetic_moment / volume);  // vector
 QUANTITY_SPEC(magnetic_field_strength, magnetization);   // vector
 inline constexpr auto magnetizing_field = magnetic_field_strength;
-QUANTITY_SPEC(permeability, magnetic_flux_density / magnetic_field_strength, quantity_character::scalar);
+// QUANTITY_SPEC(permeability, magnetic_flux_density / magnetic_field_strength, quantity_character::scalar);
 QUANTITY_SPEC(relative_permeability, dimensionless, permeability / magnetic_constant);
 QUANTITY_SPEC(magnetic_susceptibility, dimensionless, magnetization / magnetic_field_strength,
               quantity_character::scalar);
@@ -101,15 +101,15 @@ QUANTITY_SPEC(current_linkage, electric_current);
 QUANTITY_SPEC(number_of_turns_in_a_winding, dimensionless);
 QUANTITY_SPEC(reluctance, magnetic_tension / magnetic_flux);
 QUANTITY_SPEC(permeance, 1 / reluctance);
-QUANTITY_SPEC(inductance, linked_flux / electric_current);
-inline constexpr auto self_inductance = inductance;
-QUANTITY_SPEC(mutual_inductance, linked_flux / electric_current);
+// QUANTITY_SPEC(inductance, linked_flux / electric_current);
+// inline constexpr auto self_inductance = inductance;
+// QUANTITY_SPEC(mutual_inductance, linked_flux / electric_current);
 QUANTITY_SPEC(coupling_factor, dimensionless, mutual_inductance / pow<1, 2>(pow<2>(self_inductance)));
 QUANTITY_SPEC(leakage_factor, dimensionless, pow<2>(coupling_factor));
 QUANTITY_SPEC(conductivity, electric_current_density / electric_field_strength, quantity_character::scalar);
 QUANTITY_SPEC(resistivity, 1 / conductivity);
-// QUANTITY_SPEC(power, voltage* electric_current);  // TODO conflicts with mechanical power
-// inline constexpr auto instantaneous_power = power;
+QUANTITY_SPEC(electric_power, power, voltage* electric_current);
+inline constexpr auto instantaneous_power = electric_power;
 QUANTITY_SPEC(instantaneous_power, voltage* electric_current);
 QUANTITY_SPEC(resistance, voltage / electric_current);
 QUANTITY_SPEC(conductance, 1 / resistance);
diff --git a/src/systems/isq/include/mp-units/systems/isq/mechanics.h b/src/systems/isq/include/mp-units/systems/isq/mechanics.h
index 2e9bda622..b68987484 100644
--- a/src/systems/isq/include/mp-units/systems/isq/mechanics.h
+++ b/src/systems/isq/include/mp-units/systems/isq/mechanics.h
@@ -47,51 +47,55 @@ inline constexpr auto dynamic_friction_force = kinetic_friction_force;
 QUANTITY_SPEC(rolling_resistance, force);  // vector
 inline constexpr auto rolling_drag = rolling_resistance;
 inline constexpr auto rolling_friction_force = rolling_resistance;
-QUANTITY_SPEC(drag_force, force);                            // vector
-QUANTITY_SPEC(impulse, force* time);                         // vector
-QUANTITY_SPEC(angular_momentum, position_vector* momentum);  // vector
-QUANTITY_SPEC(moment_of_inertia, angular_momentum / angular_velocity, quantity_character::tensor);
-QUANTITY_SPEC(moment_of_force, position_vector* force);  // vector
+QUANTITY_SPEC(drag_force, force);                                           // vector
+QUANTITY_SPEC(impulse, force* time);                                        // vector
+QUANTITY_SPEC(angular_momentum, cross_product(position_vector, momentum));  // vector
+// QUANTITY_SPEC(moment_of_inertia, tensor_product(angular_momentum, angular_velocity), quantity_character::tensor);
+QUANTITY_SPEC(moment_of_force, cross_product(position_vector, force));  // vector
 QUANTITY_SPEC(torque, moment_of_force, quantity_character::scalar);
 QUANTITY_SPEC(angular_impulse, moment_of_force* time);  // vector
 QUANTITY_SPEC(pressure, force / area, quantity_character::scalar);
 QUANTITY_SPEC(gauge_pressure, pressure);
-QUANTITY_SPEC(stress, pressure, quantity_character::tensor);
-QUANTITY_SPEC(normal_stress, pressure, quantity_character::scalar);
-QUANTITY_SPEC(shear_stress, pressure, quantity_character::scalar);
-QUANTITY_SPEC(strain, dimensionless, quantity_character::tensor);
+QUANTITY_SPEC(stress, pressure, quantity_character::tensor);  // TODO what is a correct equation here?
+// QUANTITY_SPEC(normal_stress, pressure, quantity_character::scalar);
+// QUANTITY_SPEC(shear_stress, pressure, quantity_character::scalar);
+// QUANTITY_SPEC(strain, dimensionless, quantity_character::tensor);
 QUANTITY_SPEC(relative_linear_strain, length / length);
-QUANTITY_SPEC(shear_strain, dimensionless, displacement / thickness, quantity_character::scalar);
+// QUANTITY_SPEC(shear_strain, dimensionless, displacement / thickness, quantity_character::scalar);
 QUANTITY_SPEC(relative_volume_strain, volume / volume);
 QUANTITY_SPEC(Poisson_number, dimensionless, width / length);
-QUANTITY_SPEC(modulus_of_elasticity, normal_stress / relative_linear_strain);
-inline constexpr auto Young_modulus = modulus_of_elasticity;
-QUANTITY_SPEC(modulus_of_rigidity, shear_stress / shear_strain);
-inline constexpr auto shear_modulus = modulus_of_rigidity;
+// QUANTITY_SPEC(modulus_of_elasticity, normal_stress / relative_linear_strain);
+// inline constexpr auto Young_modulus = modulus_of_elasticity;
+// QUANTITY_SPEC(modulus_of_rigidity, shear_stress / shear_strain);
+// inline constexpr auto shear_modulus = modulus_of_rigidity;
 QUANTITY_SPEC(modulus_of_compression, pressure / relative_volume_strain);
 inline constexpr auto bulk_modulus = modulus_of_compression;
 QUANTITY_SPEC(compressibility, 1 / volume * (volume / pressure));
 QUANTITY_SPEC(second_axial_moment_of_area, pow<2>(radial_distance) * area);
 QUANTITY_SPEC(second_polar_moment_of_area, pow<2>(radial_distance) * area);
 QUANTITY_SPEC(section_modulus, second_axial_moment_of_area / radial_distance);
-QUANTITY_SPEC(static_friction_coefficient, dimensionless, static_friction_force / force, quantity_character::scalar);
+QUANTITY_SPEC(static_friction_coefficient, dimensionless, norm(static_friction_force) / norm(force),
+              quantity_character::scalar);
 inline constexpr auto static_friction_factor = static_friction_coefficient;
 inline constexpr auto coefficient_of_static_friction = static_friction_coefficient;
-QUANTITY_SPEC(kinetic_friction_factor, dimensionless, kinetic_friction_force / force, quantity_character::scalar);
+QUANTITY_SPEC(kinetic_friction_factor, dimensionless, norm(kinetic_friction_force) / norm(force),
+              quantity_character::scalar);
 inline constexpr auto dynamic_friction_factor = kinetic_friction_factor;
-QUANTITY_SPEC(rolling_resistance_factor, force / force, quantity_character::scalar);
-QUANTITY_SPEC(drag_coefficient, dimensionless, drag_force / (mass_density * pow<2>(speed) * area),
+QUANTITY_SPEC(rolling_resistance_factor, norm(force) / norm(force), quantity_character::scalar);
+QUANTITY_SPEC(drag_coefficient, dimensionless, norm(drag_force) / (mass_density * pow<2>(speed) * area),
               quantity_character::scalar);
 inline constexpr auto drag_factor = drag_coefficient;
-QUANTITY_SPEC(dynamic_viscosity, shear_stress* length / velocity, quantity_character::scalar);
-QUANTITY_SPEC(kinematic_viscosity, dynamic_viscosity / mass_density);
-QUANTITY_SPEC(surface_tension, force / length, quantity_character::scalar);  // TODO what is a correct equation here?
-QUANTITY_SPEC(power, force* velocity, quantity_character::scalar);
+// QUANTITY_SPEC(dynamic_viscosity, shear_stress* length / velocity, quantity_character::scalar);
+// QUANTITY_SPEC(kinematic_viscosity, dynamic_viscosity / mass_density);
+QUANTITY_SPEC(surface_tension, nrom(force) / length, quantity_character::scalar);
+QUANTITY_SPEC(power, mass* pow<2>(length) / pow<3>(time));  // differs from ISO 80000
+QUANTITY_SPEC(mechanical_power, power, dot_product(force, velocity),
+              quantity_character::scalar);                              // differs from ISO 80000
 QUANTITY_SPEC(energy, mass* pow<2>(length) / pow<2>(time));             // ISO 80000 defines this in thermodynamics
 QUANTITY_SPEC(mechanical_energy, energy);                               // differs from ISO 80000
 QUANTITY_SPEC(potential_energy, mechanical_energy);                     // differs from ISO 80000
 QUANTITY_SPEC(kinetic_energy, mechanical_energy, mass* pow<2>(speed));  // differs from ISO 80000
-QUANTITY_SPEC(mechanical_work, force* displacement, quantity_character::scalar);
+QUANTITY_SPEC(mechanical_work, dot_product(force, displacement), quantity_character::scalar);
 inline constexpr auto work = mechanical_work;
 QUANTITY_SPEC(efficiency_mechanics, power / power);
 QUANTITY_SPEC(mass_flow, mass_density* velocity);  // vector
diff --git a/test/unit_test/runtime/linear_algebra_test.cpp b/test/unit_test/runtime/linear_algebra_test.cpp
index b0ee38f88..5b717e6bd 100644
--- a/test/unit_test/runtime/linear_algebra_test.cpp
+++ b/test/unit_test/runtime/linear_algebra_test.cpp
@@ -35,9 +35,6 @@
 template<typename Rep = double>
 using vector = STD_LA::fixed_size_column_vector<Rep, 3>;
 
-template<typename Rep>
-inline constexpr bool mp_units::is_vector<vector<Rep>> = true;
-
 namespace STD_LA {
 
 template<typename Rep>
@@ -51,35 +48,58 @@ std::ostream& operator<<(std::ostream& os, const ::vector<Rep>& v)
   return os;
 }
 
-}  // namespace STD_LA
+template<typename Rep1, typename Rep2, auto N>
+[[nodiscard]] decltype(Rep1{} * Rep2{}) dot(const fixed_size_column_vector<Rep1, N>& a,
+                                            const fixed_size_column_vector<Rep2, N>& b)
+{
+  return 42;
+}
 
-namespace {
+template<typename Rep1, typename Rep2, auto N>
+[[nodiscard]] fixed_size_column_vector<decltype(Rep1{} * Rep2{}), N> cross(const fixed_size_column_vector<Rep1, N>& a,
+                                                                           const fixed_size_column_vector<Rep2, N>& b)
+{
+  return {a(1) * b(2) - a(2) * b(1), a(2) * b(0) - a(0) * b(2), a(0) * b(1) - a(1) * b(0)};
+}
 
-using namespace mp_units;
-using namespace mp_units::si::unit_symbols;
+namespace detail {
 
-template<typename T>
-[[nodiscard]] auto get_magnitude(const vector<T>& v)
+template<typename Rep, auto N, std::size_t... I>
+[[nodiscard]] auto norm_impl(const fixed_size_column_vector<Rep, N>& v, std::index_sequence<I...>)
 {
   using namespace std;
-  return hypot(v(0), v(1), v(2));
+  return sqrt((... + (v(I) * v(I))));
 }
 
-template<typename T, typename U>
-[[nodiscard]] vector<decltype(T{} * U{})> cross_product(const vector<T>& a, const vector<U>& b)
+}  // namespace detail
+
+template<typename Rep, auto N>
+[[nodiscard]] auto norm(const fixed_size_column_vector<Rep, N>& v)
 {
-  return {a(1) * b(2) - a(2) * b(1), a(2) * b(0) - a(0) * b(2), a(0) * b(1) - a(1) * b(0)};
+  using namespace std;
+  if constexpr (N == 1)
+    return v(0);
+  else if constexpr (N == 2)
+    return hypot(v(0), v(1));
+  else if constexpr (N == 3)
+    return hypot(v(0), v(1), v(2));
+  else
+    return detail::norm_impl(v, std::make_index_sequence<N>{});
 }
 
-template<Quantity Q1, Quantity Q2>
-  requires is_vector<typename Q1::rep> && is_vector<typename Q2::rep> &&
-           requires(typename Q1::rep v1, typename Q2::rep v2) { cross_product(v1, v2); }
-[[nodiscard]] QuantityOf<Q1::quantity_spec * Q2::quantity_spec> auto cross_product(const Q1& q1, const Q2& q2)
+}  // namespace STD_LA
+
+template<typename Rep>
+inline constexpr bool mp_units::is_vector<vector<Rep>> = true;
+
+template<typename Rep, mp_units::Reference R>
+[[nodiscard]] constexpr mp_units::quantity<R{}, vector<Rep>> operator*(const vector<Rep>& lhs, R)
 {
-  return cross_product(q1.number(), q2.number()) * (Q1::reference * Q2::reference);
+  return mp_units::make_quantity<R{}>(lhs);
 }
 
-}  // namespace
+using namespace mp_units;
+using namespace mp_units::si::unit_symbols;
 
 TEST_CASE("vector quantity", "[la]")
 {
@@ -101,7 +121,7 @@ TEST_CASE("vector quantity", "[la]")
   SECTION("to scalar magnitude")
   {
     const auto v = vector<int>{2, 3, 6} * isq::velocity[km / h];
-    const auto speed = get_magnitude(v.number()) * isq::speed[v.unit];  // TODO can we do better here?
+    const quantity<isq::speed[v.unit], int> speed = norm(v);
     CHECK(speed.number() == 7);
   }
 
@@ -122,12 +142,29 @@ TEST_CASE("vector quantity", "[la]")
     }
   }
 
-  SECTION("divide by scalar value")
+  SECTION("multiply by dimensionless quantity")
+  {
+    const auto v = vector<int>{1, 2, 3} * isq::position_vector[m];
+
+    SECTION("integral")
+    {
+      SECTION("scalar on LHS") { CHECK(((2 * one) * v).number() == vector<int>{2, 4, 6}); }
+      SECTION("scalar on RHS") { CHECK((v * (2 * one)).number() == vector<int>{2, 4, 6}); }
+    }
+
+    SECTION("floating-point")
+    {
+      SECTION("scalar on LHS") { CHECK(((0.5 * one) * v).number() == vector<double>{0.5, 1., 1.5}); }
+      SECTION("scalar on RHS") { CHECK((v * (0.5 * one)).number() == vector<double>{0.5, 1., 1.5}); }
+    }
+  }
+
+  SECTION("divide by dimensionless quantity")
   {
     const auto v = vector<int>{2, 4, 6} * isq::position_vector[m];
 
-    SECTION("integral") { CHECK((v / 2).number() == vector<int>{1, 2, 3}); }
-    SECTION("floating-point") { CHECK((v / 0.5).number() == vector<double>{4., 8., 12.}); }
+    SECTION("integral") { CHECK((v / (2 * one)).number() == vector<int>{1, 2, 3}); }
+    SECTION("floating-point") { CHECK((v / (0.5 * one)).number() == vector<double>{4., 8., 12.}); }
   }
 
   SECTION("add")
@@ -273,7 +310,7 @@ TEST_CASE("vector quantity", "[la]")
     const auto r = vector<int>{3, 0, 0} * isq::position_vector[m];
     const auto f = vector<int>{0, 10, 0} * isq::force[N];
 
-    CHECK(cross_product(r, f) == vector<int>{0, 0, 30} * isq::moment_of_force[N * m]);
+    CHECK(cross(r, f) == vector<int>{0, 0, 30} * isq::moment_of_force[N * m]);
   }
 }
 
@@ -299,7 +336,7 @@ TEST_CASE("vector of quantities", "[la]")
   SECTION("to scalar magnitude")
   {
     const vector<quantity<isq::velocity[km / h], int>> v = {2 * (km / h), 3 * (km / h), 6 * (km / h)};
-    const auto speed = get_magnitude(v).number() * isq::speed[v(0).unit];  // TODO can we do better here?
+    const auto speed = norm(v).number() * isq::speed[v(0).unit];  // TODO can we do better here?
     CHECK(speed.number() == 7);
   }
 
@@ -335,6 +372,38 @@ TEST_CASE("vector of quantities", "[la]")
     }
   }
 
+  SECTION("multiply by dimensionless quantity")
+  {
+    const vector<quantity<isq::position_vector[m], int>> v = {1 * m, 2 * m, 3 * m};
+
+    SECTION("integral")
+    {
+      const vector<quantity<isq::position_vector[m], int>> result = {2 * m, 4 * m, 6 * m};
+
+      SECTION("scalar on LHS") { CHECK((2 * one) * v == result); }
+      SECTION("scalar on RHS") { CHECK(v * (2 * one) == result); }
+    }
+
+    SECTION("floating-point")
+    {
+      const vector<quantity<isq::position_vector[m], double>> result = {0.5 * m, 1. * m, 1.5 * m};
+
+      SECTION("scalar on LHS") { CHECK((0.5 * one) * v == result); }
+      SECTION("scalar on RHS") { CHECK(v * (0.5 * one) == result); }
+    }
+  }
+
+  SECTION("divide by dimensionless quantity")
+  {
+    const vector<quantity<isq::position_vector[m], int>> v = {2 * m, 4 * m, 6 * m};
+
+    SECTION("integral") { CHECK(v / (2 * one) == vector<quantity<isq::position_vector[m], int>>{1 * m, 2 * m, 3 * m}); }
+    SECTION("floating-point")
+    {
+      CHECK(v / (0.5 * one) == vector<quantity<isq::position_vector[m], double>>{4. * m, 8. * m, 12. * m});
+    }
+  }
+
   SECTION("add")
   {
     const vector<quantity<isq::position_vector[m], int>> v = {1 * m, 2 * m, 3 * m};
diff --git a/test/unit_test/static/concepts_test.cpp b/test/unit_test/static/concepts_test.cpp
index a7b0a6da0..c57e266b3 100644
--- a/test/unit_test/static/concepts_test.cpp
+++ b/test/unit_test/static/concepts_test.cpp
@@ -295,7 +295,7 @@ static_assert(!ReferenceOf<std::remove_const_t<decltype(dimensionless[one])>, is
 // Representation
 static_assert(Representation<int>);
 static_assert(Representation<double>);
-static_assert(Representation<std::complex<double>>);
+static_assert(!Representation<std::complex<double>>);
 static_assert(!Representation<bool>);
 static_assert(!Representation<std::optional<int>>);
 static_assert(!Representation<std::chrono::seconds>);
@@ -304,7 +304,7 @@ static_assert(!Representation<std::string>);
 // RepresentationOf
 static_assert(RepresentationOf<int, quantity_character::scalar>);
 static_assert(RepresentationOf<double, quantity_character::scalar>);
-static_assert(RepresentationOf<std::complex<double>, quantity_character::scalar>);
+static_assert(!RepresentationOf<std::complex<double>, quantity_character::scalar>);
 static_assert(!RepresentationOf<bool, quantity_character::scalar>);
 static_assert(!RepresentationOf<std::optional<int>, quantity_character::scalar>);
 static_assert(!RepresentationOf<std::chrono::seconds, quantity_character::scalar>);