Vectors and Dot Products 6.4 Vectors and Dot Products The Definition of the Dot Product of Two Vectors The dot product of u = and v = is Ex.’s Find each dot product.

Properties of the Dot Product Let u, v, and w be vectors in the plane or in space and let c be a scalar.

Let Find First, find u . v Find u . 2v = 2(u . v) = 2(-14) = -28

The Angle Between Two Vectors If is the angle between two nonzero vectors u and v, then Find the angle between

Definition of Orthogonal Vectors (90 degree angles) The vectors u and v are orthogonal if u . v = 0 Are the vectors orthogonal? Find the dot product of the two vectors. Because the dot product is 0, the two vectors are orthogonal. End of notes.

Finding Vector Components Let u and v be nonzero vectors such that u = w1 + w2 where w1 and w2 are orthogonal and w1 is parallel to (or a scalar multiple of) v. The vectors w1 and w2 are called vector components of u. The vector w1 is the projection of u onto v and is denoted by w1 = projvu. The vector w2 is given by w2 = u - w1. is obtuse is acute u u w2 w2 v v w1 w1

Projection of u onto v Let u and v be nonzero vectors. The projection of u onto v is

Find the projection of onto Then write u as the the sum of two orthogonal vectors, one which is projvu. w1 = projvu = w2 = u - w1 = So, u = w1 + w2 =