4. Components and Projections

We often want to break vectors up into pieces or find out how much of a vector points in a particular direction. Let's do it!

Components

In single variable calculus, we considered projectile motion where we were given vertical and horizontal velocities. In reality, we usually just know a velocity, and we need to break it up into its vertical and horizontal parts. You will do this very frequently in Physics this semester!

Trigonometry is the trick here, and it's all about SOHCAHTOA.

components

Photo from https://physicscatalyst.com/article/components-of-a-vector/

In the picture, we have a vector and want to find the x-component and y-component of the vector (such that the $\large \vec{A_x}+\vec{A_y}=\vec{A}$). Using trig, we have

$$\large \sin(\theta)=\frac{|\vec{A}_y|}{|\vec{A}|}$$

Rearranging, we have $\large |\vec{A}_y|=|\vec{A}|\sin(\theta)$. We can similarly find the length of the x-component $\large \vec{A}_x$. We have:

$$\large \begin{split} |\vec{A_y}|&=|\vec{A}|\sin(\theta)\\ |\vec{A_x}|&=|\vec{A}|\cos(\theta)\\ \end{split} $$

Here's a video where we find the magnitude of the horizontal and vertical components of a velocity given the initial velocity and angle:

Projections

Finding components is actually a specific case of finding a projection of a vector onto another vector:

projection

Photo from https://math.stackexchange.com/questions/2143370/why-would-length-of-vector-projection-be-different-from-length-of-a-vector

The projection of a vector $\large \vec{u}$ onto a vector $\large \vec{v}$ can be thought of as the "shadow" of the first vector onto the second. The formula for a vector projection is:

$$\large \text{Proj}_{\vec{v}}\vec{u} = \frac{\vec{u}\cdot\vec{v}}{|\vec{v}|^2}\vec{v}$$

If we think of the x-component and y-component of a vector as their projections onto the x and y axes, respectively, then the vector projection formula above gives the component formulas from earlier.

Here's a video where we find the projection of one vector onto another: