Vector Math

Being able to write out vectors is a useful first step, but now we need to be able to do something with those vectors.

Vector Addition

When working with vectors, it is often necessary to add together different vectors to identify a resultant vectorⓘ.

Adding vectors together is the simplest aspect of vector math. We actually did a bit of vector addition in the previous section. By defining one vector by its components, we are actually defining that vector as a sum of each component vector.

The basic idea of adding vectors is to align them “head to tail” and then draw a vector from the tail of the first vector to the tip (or head) of the last. This method is shown in Gallery 1.2.1.

In a fixed coordinate system, add vectors by adding their signed components: (A + B)ₓ = Aₓ + Bₓ and (A + B)ᵧ = Aᵧ + Bᵧ. Negative components already represent the opposite direction.

Vector addition gets more complicated when introducing vectors that have an awkward angle between them. To add together those vectors, one ‘brute force’ approach is to identify the components of each vector, and then add each component vector.

As that approach is fairly intense, it is only recommended for cases where the vectors are given in their component forms. If the vectors are given in the form of a magnitude and direction, then it is potentially easier to use the laws of sines and cosines.

Figure 1.2.1 is a reference to the variables in the following equations. The important thing to note here is the location of each greek letter in comparison to is associated latin character ($A$, $\alpha$; $B$, $\beta$; $C$, $\gamma$). Each angle corresponds to the side of the triangle opposite to it. That consistency is needed to successfully use these formulas.

$C^2=A^2+B^2-2AB\cos \gamma$ (Equation 1.2.1)​
$\frac{A}{\sin \alpha }=\frac{B}{\sin \beta }=\frac{C}{\sin \gamma }$ (Equation 1.2.2)​
The first equation, the law of cosines will be the go-to equation for identifying the magnitude of the resultant vector.

The second equation, the law of sines, is used to identify the angles between the resultant vector and the original vector, for identifying the resultant vector’s direction.

These equations are together used to solve for the resultant vector, and this approach is one of the easiest ways of dealing with vector addition where the vector magnitudes and directions are given.

Example Problem 1.2.1

A man is trying to cross a river that is flowing from west to east at $5\text{m}/\text{s}$ . The man has a velocity of $3\text{m}/\text{s}$ going north across the river. What is the man’s resulting velocity relative to the ground?

Vector Subtraction

Vector subtraction can be done just like vector addition so it is still a resultant vector, with the only difference being that one vector is reversed, as that is what happens when a vector is multiplied by $-1$. This means that you take the given vector, swap the tip and the tail, and then add them normally. Even for subtraction, vectors are always aligned tip to tail, because it is still addition, just with one vector reversed .

Vector Multiplication part 1 - Scalar Multiplication

Sometimes a scalar is multiplied by a vector. Say you wanted to find out what you would get if you doubled your velocity.

That would look like:

$2\cdot \overrightarrow{v}=?$

Multiplying a vector by a scalar changes its magnitude by the absolute value of that scalar. A positive scalar preserves its direction; a negative scalar reverses it; multiplying by zero gives the zero vector.

$2\cdot \left(4\text{m}/\text{s at }{40}^{\circ }\right)=8\text{m}/\text{s at }{40}^{\circ }$

$2\cdot ⟨\mathrm{3,4}⟩=⟨\mathrm{6,8}⟩$

$2\cdot \left(6\hat{i}+9\hat{j}\right)=12\hat{i}+18\hat{j}$

Vector Multiplication part 2 - the Dot Product

Vector multiplication with other vectors occurs in two forms, the dot product and the cross product. The dot product of vectors $\overrightarrow{s}$ and $\overrightarrow{t}$ is depicted as $\overrightarrow{s}•\overrightarrow{t}$, and their cross product is depicted as $\overrightarrow{s}\times \overrightarrow{t}$.

The dot product can be calculated in two ways: by magnitude or by components. The first equation is the magnitude equation, with $\theta$ being the angle between the two vectors:

$\overrightarrow{a}•\overrightarrow{b}=\left|\begin{array}{c}\overrightarrow{a}\end{array}\right|\left|\begin{array}{c}\overrightarrow{b}\end{array}\right|\cos \theta$ (Equation 1.2.3)​

Next is the component equation:

$\overrightarrow{a}•\overrightarrow{b}={\overrightarrow{a}}_x{\overrightarrow{b}}_x+{\overrightarrow{a}}_y{\overrightarrow{b}}_y+{\overrightarrow{a}}_z{\overrightarrow{b}}_z$ (Equation 1.2.4)​

Key Point: Dot products are scalars, and have no direction. Therefore, the order the vectors are multiplied does not matter, so $\overrightarrow{b}•\overrightarrow{a}=\overrightarrow{a}•\overrightarrow{b}$.

Vector Multiplication part 3 - the Cross Product

The goal of a cross product is to identify a vector that is perpendicular to the two vectors being multiplied, and which has a magnitude equal to the area of a parallelogram made by the vectors, as shown below.

The calculation of a cross product is fairly simple if the magnitudes and directions of the vectors in question are known. The cross product is found via the following formula:

$\overrightarrow{a}\times \overrightarrow{b}=\left(\left|\begin{array}{c}\overrightarrow{a}\end{array}\right|\left|\begin{array}{c}\overrightarrow{b}\end{array}\right|\sin \theta \right)\hat{u}$ (Equation 1.2.5)​
The unit vector $\hat{u}$ in the equation is a unit vector whose direction is determined via the right-hand ruleⓘ. This rule is shown below.

Key Point: A nonzero cross product is perpendicular to both input vectors. If the vectors are parallel or either is zero, their cross product is the zero vector and has no direction.

Example Problem 1.2.2

What is the cross product, $\overrightarrow{a}\times \overrightarrow{b}$, if the vectors are defined as:

$\overrightarrow{a}=5\text{m}/\text{s North}$​
$\overrightarrow{b}=10\text{m at }{45}^{\circ }\text{ N of E}$