Projection of Vectors

Video

(Video) Vector Projections

by Firefly Lectures

In this video by Firefly Lectures, you'll learn about what vector projections are and their applications.

Summary

We can think of a vector projection as a shadow of one vector onto another. From the video, the green projection vector is the projection of $\vec{u}$ onto $\vec{v}$ . When we are looking at projections, we are looking at scalar quantities.

There's a formula for the projection of a vector:

\text{proj}_{\vec{v}} \vec{u} = \frac{\vec{u} \cdot \vec{v}} {||\vec{v}||^2} \vec{v}

It's important to note here that the dot product is scalar, which allows the vector projection to be scalar.

Interactive · 1 question

By Anusha Rahman

Now that we've already learned about vectors more generally, including dot products, operations, magnitude and unit vectors, and cross products, we can now dive into the projection of vectors.

In simple terms, the projection of a vector is like "casting" one vector onto another. Imagine shining a light onto a vector so it creates a shadow on another vector. The shadow is the projection.

Vector Projections

Projections Helps us to understand how one vector "acts" along the direction of another vector. This concept is useful in many areas of physics and engineering. For example, if you're moving along a slope, the projection of your velocity on the direction of the slope shows how fast you are moving along the shape.

Suppose you have two vectors, vector A and vector B. You want to "project" vector A onto vector B. You want to know what the "shadow" of vector A would look like.

This idea seems complicated - but there is a formula that can help us! The projection of a vector A onto a vector B is a vector that points in the direction of B and has a length that represents how much of A is in the direction of B.

\text{proj}_B A = \frac{A \cdot B}{B \cdot B} B

In this formula, remember that:

$A \cdot B$ is the dot product of vectors A and B.
$B \cdot B$ is the dot product of vector B with itself, which is just the magnitude of B squared.
$\text{proj}_{xx}$ represents the projection of A onto B.

Example

Given:

Vector A = $\langle 3, 4 \rangle$
Vector B = $\langle 1,2 \rangle$

We want to find the projection of A onto B.

Step 1: Calculate the Dot Product of $A \cdot B$

The dot product of two vectors $A \cdot B = a_1b_1 + a_2b_2$

So for our vectors A and B:

A \cdot B = (3)(1) + (4)(2) = 11

Step 2: Calculate $B \cdot B$

Now we need to calculate the dot product of B with itself:

B \cdot B = (1)(1) + (2)(2) = 5

Step 3: Apply the Formula for the Projection

Now, we can use the projection formula: $\text{proj}_B A = \frac{A \cdot B}{B \cdot B} B$

\text{proj}_B A = \frac{11}{5} \langle 1,2 \rangle

Now, multiply the scalar $\frac{11}{5}$ by each component of vector B;

\text{proj}_B A = \langle \frac{11}{5}, \frac{22}{5} \rangle

We've found the projection of vector A onto vector B!

#####Interpretation

The projection vector $\langle \frac{11}{5}, \frac{22}{5} \rangle$ is a vector that points in the direction of B and has the length that represents how much of A lies in the direction of B. If you imagine vector B as a straight line, the projection vector is the shadow of A on that line.

Some Important Notes

The projection will always be a vector that points along B. If A and B are in the same direction, the projection will be exactly A. If they are at a right angle, then the projection will be the zero vector, meaning that there's no "shadow" or component of A in the direction of B.

The second big thing to note is that the length of the projection depends on how much A "overlaps" with B. If A and B are pointing in very different directions, the projection will be shorter.

Another Example

Here's another example for you to try on your own.

Find the projection of vector A $= \langle 4, -3 \rangle$ onto vector B $= \langle 2,1 \rangle$

Are you ready to see the answer?

Illustrated · 1 min

By Anusha Rahman