Calculus 10.3 (The Dot Product)

Finding the dot product of two vectors offers a method of determining the angle between the vectors or the projection of one vector on another using the “definition of the dot product” theorem

To find the dot product of two vectors, the corresponding components are multiplied and the products of each are added to obtain the scalar that is the dot product

“Scalar product” or “Inner product”

The dot product is fundamentally a projection. As shown in this image, the dot product of a vector with a unit vector is the projection of that vector in the direction given by the unit vector

Use the theorem of the definition of the dot product to find the scalar result of: 12

The result of dotting a vector with itself is its length squared. This is because the angle between a vector and itself is 0 and the Cos(0) = 1
This is derived directly from the definition of the dot product theorem

The result of dotting a vector with its inverse is its length squared since length is a scalar void of direction. This is because the angle between a vector and its inverse is 180 and Cos(180) = 1
This is derived directly from the definition of the dot product theorem

The result of dotting a vector with itself is its length squared, and since a unit vector has a length of 1 the dot product will also be equal to 1

Two vectors are perpendicular or orthogonal only if the result of their dot product is 0. This is because the the Cos(90) = 0
This is derived directly from the definition of the dot product theorem

The components of a vector are just the dot products of the vector with each basis vector (i,j,k)

If it is considered that a vectors components are each a product of the vector with a basis vector (i,j,k). The equation for the dot product of two vectors can be proofed by the FOIL method and the knowledge that the dot product of a unit vector with itself equals 1 while the dot product of orthogonal vectors equals 0

Using vector addition of the triangle law, the knowledge that a vector dotted with itself equals its magnitude squared, and the definition of the dot product theorem it is possible to see where the law of cosines is derived from!

Beginning with three independent vectors, choose one (Black) as the first of the new orthogonal basis vectors. Then pick a second (thick red), and subtract from this vector its projection parallel to the first, resulting in a vector perpendicular to the first (thin red). Now subtract from the remaining vector (thick blue) its projections parallel to both the first and second vectors, resulting in a vector perpendicular to both (thinnest blue)

True, dot products are distributive

The dot product of a vector and a scalar simply distributes the scalar into each component of the vector resulting in the vector of changed length sustaining its direction and remaining a vector

The dot product of two vectors is a scalar. Using either the definition of the dot product theorem or the equation of a dot product gives either an angle or magnitude, both of which are void of direction

The scalar projection of one vector onto another vector signifies the component of magnitude that the first vector has in the same direction as the second vector. This magnitude is solely a scalar quantity without direction

The vector projection of one vector onto another vector signifies a new vector whose magnitude is equal to the component of magnitude that the first vector has in the same direction as the second vector and whose direction is the same as the second vector

W = Fd
The work done by a constant force F is the the dot product of
F and D, where D is the displacement vector. Since force and displacement are vectors with direction, the work done is a magnitude scalar without direction.

Use the equation for work along with the definition of the dot product

Scalar; work is calculated by the equation
W = Fd and since force and displacement are vectors and so their dot product must be a magnitude scalar

Use the equation for work along with the definition of the dot product and the equation for vector displacement.

Use the definition of the dot product to determine the scalar projection of b onto a and then multiply that magnitude by the unit vector in the direction of vector a in order to determine the vector projection of b onto a

These vectors are orthogonal. The definition of the dot product specifies that if the dot product of two vectors is equal to the product of their magnitudes the vectors are parallel or if the dot product is 0 then the vectors are orthogonal. In this case the dot product is 0 and so the vectors are orthogonal since cos(90) = 0