Prep Precalculus D 2006-07: 10-4: Scalar Products and Projections of Vectors

10-4 Scalar Products and Projections of Vectors

The Objective: In a situation where you are given two vectors, find their dot product. Use the result to find the angle between the vectors and the projection of one vector on the other.

Basically this chapter is dealing with multiplying vectors together using various methods and answering various questions.

*An important thing to remember* is that dot products, scalar products, and inner products are all the same thing.

The equation for the DOT PRODUCT of vector a and vector b is vector "a" times vector "b" or = a dot b (both are vectors)

If you translate the vectors so that they are tail-to-tail you find the dot product by multiplying the magnitudes of the vectors and the cosine of the angle between them

For example (vectors "v" and "u"):

The definition of a dot product is vector "a" times vector "b" equals the magnitude of vector "a" times the magnitude of vector "b" times the cosine of the included angle.

Technique for the computation of dot product:

vector a= X1i + Y1j + Z1k
and
vector b= X2i + Y2j + Z2k
then

a dot b = X1X2 + Y1Y2 + Z1Z2

Maybe it will help to clarify in verbal terms: The dot product of two three-dimensional vectors equals the sum of the respective products of the coefficients for the i, j, and k unit vectors.

Example

Find the dot product of vectors a and b.

vector a= 3i - 4j + 12k
vector b= 2i + 6j - 3k

a dot b = 3(2) + (-4 )( 6) + (12 ) (-3) = -54

To find the measure of the angle theta between vectors x and y you would take:

a dot b = -54

(7 ) (13) ( cos theta)= -54

cos theta= -54/ (7 ) (13 )

cos^-1(-.5934)= 126.399 degrees (theta)

The other portion of this section deals with Projections of Vectors

The component is called vector projection of vector a on vector b.

An example of this would be light rays shining on two vectors. The rays would be perpendicular to the bottom vector made up of two black arrows (forming a right angle).