Vector Refresher Part 3• Vector Dot Product

Definitions• Some Properties

• The Angle Between 2 Vectors

• Scalar Projections• Vector Projections

Dot Product• One form of vector multiplication• Yields a SCALAR quantity• Can be used to find the angle between 2

vectors• Can also be used to find the projection of a

vector in a given direction

Symbolism• The dot product is symbolized with a dot

between 2 vectors

Symbolism• The dot product is symbolized with a dot

between 2 vectors• The following means “Vector A dotted with

vector B”

One DefinitionThe dot product is defined as the sum of the product of similar components of a vector

One DefinitionThe dot product is defined as the sum of the product of similar components of a vectorIf we have the following 2 vectors:

One DefinitionThe dot product is defined as the sum of the product of similar components of a vectorIf we have the following 2 vectors:

One DefinitionThe dot product is defined as the sum of the product of similar components of a vectorIf we have the following 2 vectors:

NOTE: This is a SCALAR term whose units are the product of the units of the 2 vectors

Another DefinitionThe dot product is also related to the angle produced by arranging 2 vectors tail to tail.

Another DefinitionThe dot product is also related to the angle produced by arranging 2 vectors tail to tail.If we have the following 2 vectors:

θ

Properties of the Dot Product

Commutative:

Properties of the Dot Product

Commutative:Associative:

Properties of the Dot Product

Commutative:Associative:Distributive:

The Angle Between 2 Vectors

The dot product is a useful tool in determining the angle between 2 vectors

θ

The Angle Between 2 Vectors

The dot product is a useful tool in determining the angle between 2 vectors

θ

The Angle Between 2 Vectors

The dot product is a useful tool in determining the angle between 2 vectors

θ

The Angle Between 2 Vectors

The dot product is a useful tool in determining the angle between 2 vectors

θ

The Angle Between 2 Vectors

The dot product is a useful tool in determining the angle between 2 vectors

θ

If 2 vectors are orthogonal, their dot product is 0

Scalar ProjectionThe dot product is also used to determine how much of a vector is acting in a particular direction.

Scalar ProjectionThe dot product is also used to determine how much of a vector is acting in a particular direction.

θ

Scalar ProjectionThe dot product is also used to determine how much of a vector is acting in a particular direction.

θ

If we want to find how much of acts in the direction of , (length of the green line) we can use the dot product

Scalar ProjectionThe dot product is also used to determine how much of a vector is acting in a particular direction.

θ

If we want to find how much of acts in the direction of , (length of the green line) we can use the dot product

Scalar ProjectionThe dot product is also used to determine how much of a vector is acting in a particular direction.

θ

If we want to find how much of acts in the direction of , (length of the green line) we can use the dot product

Note that this result is a SCALAR quantity, meaning that it has no direction associated.

Scalar ProjectionThe dot product is also used to determine how much of a vector is acting in a particular direction.

θ

If we want to find how much of acts in the direction of , (length of the green line) we can use the dot product

Note that this result is a SCALAR quantity, meaning that it has no direction associated. Thus, this calculation is the scalar projection

Vector ProjectionThe scalar projection can be used to determine a vector projection

θ

We can transform the scalar projection, in this case , into a vector by multiplying the scalar projection and the unit vector that described the direction of interest, in this case

This is a VECTOR quantity that describes the vector shown by the green arrow

Applications of the Vector Projection

We can use the vector projection to determine the vector parallel and perpendicular to a given direction

θ

Applications of the Vector Projection

We can use the vector projection to determine the vector parallel and perpendicular to a given direction

θ

A vector can be described as its vector component parallel to a direction plus its component perpendicular to a direction

Applications of the Vector Projection

We can use the vector projection to determine the vector parallel and perpendicular to a given direction

θ

A vector can be described as its vector component parallel to a direction plus its component perpendicular to a direction

Example ProblemIf and , find the angle between the 2 vectors, the projection of onto , and the component of that is perpendicular to .

Example ProblemIf and , find the angle between the 2 vectors, the projection of onto , and the component of that is perpendicular to . Looking at this formula, we need

to determine the magnitude of each vector and evaluate the dot product

Example ProblemIf and , find the angle between the 2 vectors, the projection of onto , and the component of that is perpendicular to . We can start by finding the

magnitude of vector U

Example ProblemIf and , find the angle between the 2 vectors, the projection of onto , and the component of that is perpendicular to . We can start by finding the

magnitude of vector U

Example ProblemIf and , find the angle between the 2 vectors, the projection of onto , and the component of that is perpendicular to . We can start by finding the

magnitude of vector U

Example ProblemIf and , find the angle between the 2 vectors, the projection of onto , and the component of that is perpendicular to . Now, we can do the same for

vector V

Example ProblemIf and , find the angle between the 2 vectors, the projection of onto , and the component of that is perpendicular to . Now, we can do the same for

vector V

Example ProblemIf and , find the angle between the 2 vectors, the projection of onto , and the component of that is perpendicular to . Now, we can do the same for

vector V

Example ProblemIf and , find the angle between the 2 vectors, the projection of onto , and the component of that is perpendicular to . Next, we’ll take the dot product to

complete the formula.

Example ProblemIf and , find the angle between the 2 vectors, the projection of onto , and the component of that is perpendicular to . Now, we can use the inverse

cosine function to find the angle

Example ProblemIf and , find the angle between the 2 vectors, the projection of onto , and the component of that is perpendicular to . Now, we can use the inverse

cosine function to find the angle

Example ProblemIf and , find the angle between the 2 vectors, the projection of onto , and the component of that is perpendicular to . To find the projection of U onto V,

we need to use the formula to the left, which means we need the unit vector that describes the direction of V

Example ProblemIf and , find the angle between the 2 vectors, the projection of onto , and the component of that is perpendicular to . We already calculated the

magnitude of V. We’ll use that to find the unit vector

Example ProblemIf and , find the angle between the 2 vectors, the projection of onto , and the component of that is perpendicular to . We already calculated the

magnitude of V. We’ll use that to find the unit vector

Example ProblemIf and , find the angle between the 2 vectors, the projection of onto , and the component of that is perpendicular to . Now, we can take the dot product

to find the scalar projection.

Example ProblemIf and , find the angle between the 2 vectors, the projection of onto , and the component of that is perpendicular to . Now, we can take the dot product

to find the scalar projection.

Example ProblemIf and , find the angle between the 2 vectors, the projection of onto , and the component of that is perpendicular to . To find the vector projection, we’ll

apply the scalar projection to the unit vector that describes the direction of V.

Example ProblemIf and , find the angle between the 2 vectors, the projection of onto , and the component of that is perpendicular to . To find the vector projection, we’ll

apply the scalar projection to the unit vector that describes the direction of V.

Example ProblemIf and , find the angle between the 2 vectors, the projection of onto , and the component of that is perpendicular to . To find the vector projection, we’ll

apply the scalar projection to the unit vector that describes the direction of V.

Example ProblemIf and , find the angle between the 2 vectors, the projection of onto , and the component of that is perpendicular to . Finally, we can subtract the

component of U parallel to V from U to get the part of U that is perpendicular to V.

Example ProblemIf and , find the angle between the 2 vectors, the projection of onto , and the component of that is perpendicular to . Finally, we can subtract the

component of U parallel to V from U to get the part of U that is perpendicular to V.

Example ProblemIf and , find the angle between the 2 vectors, the projection of onto , and the component of that is perpendicular to . We can check our work with the

following formula

because the parallel and perpendicular components of U form a right triangle, with U as the hypotenuse.

Example ProblemIf and , find the angle between the 2 vectors, the projection of onto , and the component of that is perpendicular to . We can check our work with the

following formula

because the parallel and perpendicular components of U form a right triangle, with U as the hypotenuse.

Example ProblemIf and , find the angle between the 2 vectors, the projection of onto , and the component of that is perpendicular to . We can check our work with the

following formula

because the parallel and perpendicular components of U form a right triangle, with U as the hypotenuse.

Example ProblemIf and , find the angle between the 2 vectors, the projection of onto , and the component of that is perpendicular to . We can check our work with the

following formula

because the parallel and perpendicular components of U form a right triangle, with U as the hypotenuse.

Example ProblemIf and , find the angle between the 2 vectors, the projection of onto , and the component of that is perpendicular to . We can check our work with the

following formula

because the parallel and perpendicular components of U form a right triangle, with U as the hypotenuse.

Example ProblemIf and , find the angle between the 2 vectors, the projection of onto , and the component of that is perpendicular to . We can check our work with the

following formula

because the parallel and perpendicular components of U form a right triangle, with U as the hypotenuse.

Example ProblemIf and , find the angle between the 2 vectors, the projection of onto , and the component of that is perpendicular to . We can check our work with the

following formula

because the parallel and perpendicular components of U form a right triangle, with U as the hypotenuse.

Example ProblemIf and , find the angle between the 2 vectors, the projection of onto , and the component of that is perpendicular to . We can check our work with the

following formula

because the parallel and perpendicular components of U form a right triangle, with U as the hypotenuse.