Mastering Parametric Vectors and Planes

This guide covers all essential equations and concepts for parametric vectors as they relate to planes in multivariable calculus.

1. Parametric Equations of a Line

A line in 3D can be described parametrically as:

$$ \vec{r}(t) = \vec{r}_0 + t\vec{v} $$

• $\vec{r}_0$: Position vector of a point on the line
• $\vec{v}$: Direction vector
• $t$: Parameter

2. Vector Equation of a Plane

A plane can be described by:

$$ \vec{r} = \vec{r}_0 + s\vec{a} + t\vec{b} $$

• $\vec{r}_0$: Position vector of a point on the plane
• $\vec{a}, \vec{b}$: Non-parallel direction vectors in the plane
• $s, t$: Parameters

3. Scalar Equation of a Plane

The scalar (Cartesian) equation:

$$ Ax + By + Cz = D $$

• $(A, B, C)$: Components of the normal vector $\vec{n}$
• $(x, y, z)$: Coordinates of any point on the plane
• $D$: Constant

4. Normal Vector to a Plane

Given two vectors in the plane $\vec{a}$ and $\vec{b}$:

$$ \vec{n} = \vec{a} \times \vec{b} $$

• $\vec{n}$: Normal vector (perpendicular to the plane)

5. Distance from a Point to a Plane

For point $P(x_1, y_1, z_1)$ and plane $Ax + By + Cz = D$:

$$ \text{Distance} = \frac{|A x_1 + B y_1 + C z_1 - D|}{\sqrt{A^2 + B^2 + C^2}} $$

6. Intersection of a Line and a Plane

Substitute the parametric line into the plane equation and solve for the parameter:

$$ A(x_0 + at) + B(y_0 + bt) + C(z_0 + ct) = D $$ Solve for $t$ to find the intersection point.

7. Angle Between Two Planes

Given normals $\vec{n}_1$ and $\vec{n}_2$:

$$ \cos \theta = \frac{\vec{n}_1 \cdot \vec{n}_2}{|\vec{n}_1||\vec{n}_2|} $$

8. Projection of a Vector onto a Plane

Given vector $\vec{v}$ and plane normal $\vec{n}$:

$$ \text{Proj}_{\text{plane}}(\vec{v}) = \vec{v} - \left(\frac{\vec{v} \cdot \vec{n}}{|\vec{n}|^2}\right)\vec{n} $$

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Tip: Always identify the normal vector and a point on the plane to use these equations effectively.

Summary Table

Concept	Equation
Parametric Line	$\vec{r}(t)=\vec{r}_0+t\vec{v}$
Parametric Plane	$\vec{r}=\vec{r}_0+s\vec{a}+t\vec{b}$
Scalar Plane	$Ax+By+Cz=D$
Normal Vector	$\vec{n}=\vec{a}\times\vec{b}$
Point–Plane Distance	$D=\dfrac{\lvert ax_0+by_0+cz_0+d\rvert}{\sqrt{a^2+b^2+c^2}}$
Line–Plane Intersection	Substitute line into plane, solve for $t$
Angle Between Planes	$\cos\theta=\dfrac{\lvert \vec n_1\cdot\vec n_2\rvert}{\lVert \vec n_1\rVert,\lVert \vec n_2\rVert}$
Projection onto Plane	$\vec v_{\mathrm{plane}}=\vec v-\dfrac{\vec v\cdot\vec n}{\lVert \vec n\rVert^2},\vec n$

This covers all the equations you need to master parametric vectors and planes.