11.3 The Dot Product

Geometric interpretation of dot product

A dot product is the magnitude of one vector times the portion of the vector that points in the same direction as that vector (the projection in the direction of the other.)

Find the dot product of the given vectors

• u = (4,10) v = (-2,3)

• u = (1,5,7) v = (-1,3,4)

Note: To find a dot product on the TI89

Press 2nd 5 (math) – 4 matrix – L Vector ops – 3 dotP

dotP([1,5,7],[-1,3,4])

Note property 5 is explained on the next slide.

__We can see that using the Pythagorean Theorem yields the same result as √a∙a

So we can write the length or magnitude of a vector in terms of the dot product. This will be important in the second semester.

Angle Between Two Vectors

Proven on last slide

Alternative form of dot product

Notes: This definition will allow us to expand the notion of orthogonal to higher Dimensions. (This will be important next semester in Linear Algebra.)

Orthogonal and perpendicular are generally used interchangeably. However there is a subtle difference. Perpendicular means that two items (planes, lines segments vectors … whatever) must meet to make a 90 degree angle… However, orthogonal includes this situation plus includes the zero vector is orthogonal to all other vectors even though we could not say that it is perpendicular to all other vectors.

Find the angle between u and v

• u = (3,-1,3)

• v =(-4,0,2)

What is meant by the angle between two vectors?

Determine if the given vectors are orthogonal

• u = (3,-1,2)

• w =(1,-1,3)

Determine if the given vectors are orthogonal

• u = (3,-1,2)

• w =(1,-1,3)

u and v are not orthogonal because the dot product is not 0.

What value x will make vectors u and q orthogonal?

q = (1,-1,x)

Example 5

Find the projection of u onto v and the vector component of u orthogonal to v

• u = 3i – 5j +2k v = 7i + j -2k

Note solve this problem 3 ways:Solve with special right trianglesSolve with Trigonometry and force times distanceSolve with the dot product

"A mathematician is a device for turning coffee into theorems“ -- P. Erdos

Proof of the dot product