2. Lecture: Vectors

Vector Geometry

Geometrically, a vector is an arrow with a length and direction:

vectors_1

Vectors with the same length and direction are called equivalent:

vectors_2

Vectors in the same (or opposite) direction are called parallel:

vectors_3

We can add and subtract vectors (click on the image to see the video):

We can multiply vectors by a real number (called a scalar), and this just scales the vector:

vectors_4

All of this is true for vectors in 3D as well:

3dvector

Vector Algebra

Alegbraically, a vector is a list of numbers: $\large \vec{v}=\left<2,4,-5\right>$. The numbers are called components. In the vector $\large \vec{v}=\left<2,4,-5\right>$, the $x$-component is 2, the $y$-component is 4, and the $z$-component is $-5$. The components represent distances along the axes, so we can get back the geometric representation of the vector:

We typically use an arrow over a letter or a bold letter to denote a vector.

Addition and subtraction with components is easy: $\large \left<2,-7,0\right>+\left<1,-5,3\right>=\left<3,-12,3\right>$.

Scalar multiplication is also easy: $\large 3\left<6,-1,2\right>=\left<18,-3,6\right>$.

Vector Miscellaneous

The length of a vector, also called its magnitude is given by $\large \left|\vec{v}\right|=\left|\left<x,y,z\right>\right|=\sqrt{x^2+y^2+z^2}$.

A vector with length 1 is called a unit vector, and we use a hat notation for this $\large \hat{x}$.

The following three unit vectors are called the basis unit vectors:

$\large \hat{i}=\left<1,0,0\right>$
$\large \hat{j}=\left<0,1,0\right>$
$\large \hat{k}=\left<0,0,1\right>$

The basis vectors provide us with an alternative notation for vectors (a notation preferred by physicists!):

$$\large \left<2,3,-4\right>=2\left<1,0,0\right>+3\left<0,1,0\right>-4\left<0,0,1\right>=2\hat{i}+3\hat{j}-4\hat{k}$$

Tomorrow's class:

Practice working with vectors

Section Outcomes:

I can:

Add/subtract vectors and multiply by a scalar, both graphically and algebraically
Find the magnitude of a vector
Define the terms: component, magnitude, unit vector, parallel, equivalent vector