# Calculus III: The Dot Product (Level 8 of 12) | Scalar, Vector and Orthogonal Projections

The Dot Product level 8

In this video we will go over our first application

that makes use of the dot product. Recall

that we covered two different forms of the

dot product the component definition and geometric

definition. Having these two forms of the

dot product opens the possibility of using

basic algebraic operations like in the case

of the component definition to calculate geometric

quantities like lengths and angles via the

geometric definition.

In the previous video we used this to our

advantage to find the angle between two vectors

by rearranging the geometric definition. Another

useful application of the dot product is to

find the scalar and vector projections between

two vectors.

Many applications in physics and engineering

make use of vectors by adding two or more

vectors to produce a resultant vector. At

times the reverse problem, in this case decomposing

a given vector into the sum of two or more

vector components, is often encountered.

Given two vectors a and b with the same initial

point and theta representing the angle between

vector a and vector b, the scalar projection

of b onto a (also called the component of

vector b along vector a) is defined to be

the magnitude of the vector projection, you

can think of the scalar projection of vector

b onto vector a as the length of the shadow

that is cast when you shine a light in the

direction perpendicular or orthogonal to the

line that is parallel to vector a. Using right

triangle trigonometry we see that the scalar

projection is going to be equal to the magnitude

of vector b times cosine of theta. We mathematically

denote this length as follows, and is read

as the component of b along a. Similar to

the dot product this value will be positive

if theta is between 0 inclusive and pi over

2 exclusive, and it will be negative if theta

is between pi over 2 exclusive and pi inclusive.

Notice that this quantity appears in the geometric

definition of the dot product, using this

expression we see that the dot product between

vector a and vector b can be interpreted as

the length of vector a times the scalar projection

of vector b onto vector a.

The geometric definition of the dot product

can also be used to provide a simple way of

calculating the scalar projection of one vector

in the direction of another vector.

We know that the value of the scalar projection

is just equal to the magnitude of vector b

times cosine of theta, so we go ahead and

solve for this quantity, doing that we obtain

the following expression. Notice that the

component of vector b along vector a can be

computed by taking the dot product of vector

b with the unit vector in the direction of

vector a.

In other words, If you want to find out how

much a vector v “leans” in the direction of

a given vector you simply dot vector v with

a unit vector that points in the direction

of the given vector.

The scalar projection is nothing new, we have

already computed this quantity various times

when we broke apart a vector in R squared

into its x and y components and a vector in

R cubed when we broke it apart into its x,

y and z components. In these cases we found

the scalar projection of the vector in the

direction of the unit standard vectors i hat,

j hat in the case of R squared, and i hat,

j hat and k hat in the case of R cubed. What

makes the dot product so useful is the fact

that you can now easily calculate the component

of a vector in any direction and not just

the standard unit vectors i, j and k. The ability

to decompose a vector into its component parts

is a fundamental theme in physics, engineering

and linear algebra. We will see this in action

in a later video.

The vector projection of vector b onto vector

a (also known as the vector component of vector

b along vector a) can be computed by multiplying

the scalar projection of vector b onto vector

a with the unit vector in the direction of

vector a.

The vector projection will generate a vector

that points in the direction of the vector

that it was projected onto, with the length

of the shadow casted representing the scalar

projection. So keep this detail in mind the

scalar projection will generate a scalar that

represents the length of the shadow that is

created when one vector is projected onto

another vector, while the vector projection

will generate a vector whose magnitude is

equal to the length of the casted shadow.

In the special case where vector a is a unit

vector this expression simplifies to the following.

Personally I like to use the first form since

it makes more sense intuitively, than the

second expression, but you are free to use

either one they are essentially equivalent

expressions.

One last thing to note is that we need to

be very careful with our notation. The vector

projection of vector b onto vector a will

generate a totally different vector when compared

to the vector projection of vector a onto

vector b assuming that both vector a and vector

b are distinct vectors. In the first expression

the vector generated will be parallel to vector

a while in the second expression the vector

generated will be parallel to vector b, so

be careful with the notation and make sure

you double check that you are finding the

correct vector projection.

The final projection that you should be familiar

with is the vector component of vector b orthogonal

to vector a. This vector is denoted as follows,

this is referred to as the orthogonal projection

of vector b onto vector a, it is also known

as the vector rejection of vector b from vector

a, this vector is the orthogonal projection

of vector b onto a line that is orthogonal

to vector a. Both the vector projection and

vector rejection represent the components

of vector b. As a result the sum of the vector

rejection and vector projection is equal to

vector b. If we solve for the orthogonal projection

of vector b onto vector a it will be equal

to vector b minus the vector projection of

vector b onto vector a. Also notice that the

vector projection and vector rejection are

orthogonal to one another.

With these two projections you can essentially

break apart a vector into two components that

do not point in the direction of the standard

unit vectors i, j and k. So now you are free

to break a vector into components along any

vector. Alright in our next video we will

go over a couple of examples to illustrate

how to find these projections.

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Very well done !

Cheers