1. 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.

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

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

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

5. 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.

6. Finding Vector Components Let u and v be nonzero vectors such that u = w 1 + w 2 where w 1 and w 2 are orthogonal and w 1 is parallel to (or a scalar multiple of) v. The vectors w 1 and w 2 are called vector components of u. The vector w 1 is the projection of u onto v and is denoted by w 1 = proj v u. The vector w 2 is given by w 2 = u - w 1. w2w2 v u w1w1 is acute w1w1 u w2w2 is obtuse v

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

8. Find the projection of onto Then write u as the the sum of two orthogonal vectors, one which is proj v u. w 1 = proj v u = w 2 = u - w 1 = So, u = w 1 + w 2 =