Calculus 10.3 (The Dot Product)

The dot product is sometimes referred to as what other two names?

"Scalar product" or "Inner product"

The Gram-Schmidt Orthogonalization process: Method for determining arbitrary basis vectors given a set of linearly independent vectors

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)

Component form of dot product (proof of dot product)

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

Is work a vector or scalar?

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

If a, b, and c are vectors, a dotted with (b + c) = a dot b + a dot c (T/F)

True, dot products are distributive

How can it be proven that two vectors are perpendicular (2D) or orthogonal (3D)

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

Find the scalar projection and vector projection of vectors, b = <1,1,2> onto, a = <-2,3,1>

Use the definition of dot product theorem to derive the magnitude component of vector b in the direction of vector a. Then multiply the projection by the unit vector in the direction of vector a in order to find the vector projection

How are the components of a vector related to the dot product?

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

Dot product (equation)

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

If the vectors a and b have lengths 4 and 6, and the angle between them is 60°, find the dot product of the vectors

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

How can the Law of Cosines be derived from dot products?

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!

What is the purpose in finding the dot product of two vectors?

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

Dot product (geometric definition)

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

The dot product of a vector and a scalar is a (vector/scalar)

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 the dot product of two vectors is a (vector/scalar)

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

What is the dot product of a vector and its inverse?

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

What is the dot product of a unit vector and itself?

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

What is the dot product of a vector and itself?

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

Scalar projection (definition)

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

Vector projection (definition)

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

Are the vectors, a = 3i + 6j - 4k b = 10i - 9j - 6k orthogonal, parallel, or neither?

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

Find the angle between the vectors, a = <2,2,⁻1> b = <5,⁻3,2>

Use the definition of the dot product theorem, dot product equation, and the magnitude of a vector equation

Find the vector projection of b = 2i + 4j -1k onto a = 3i - 4j - 5k

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

A crate is hauled 8 meters up a ramp under a constant force of 200 Newtons applied at an angle of 25° to the ramp. Find the work done

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

A force is given by a vector, F = 3i + 4j + 5k and moves a particle from the point, P(2,1,0) to the point, Q(4,6,2) Find work done.

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

Equation for finding the work done by a constant force through a distance.

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.