1. 6.4 Vectors and Dot Products Objectives: Students will find the dot product of two vectors and use properties of the dot product. Students will find angles between vectors and determine whether two vectors are orthogonal. Students will write vectors as sums of two vector components.

2. Dot Product The dot product of is given by

3. Ex 1) Find the dot product of (3, -2) and (2, 5).

4. A vector is represented by coordinates ie: (-8, 3) A scalar is represented by a real number ie: -6 or 2 – √3

5. Ex 2) Let u = (-3, -2), v = (2, 1) and w = (3, -1) Find each dot product and state whether the result is a vector or a scalar. a)(u · v) w b)u · 2v

6. Ex 3) Use the dot product to find the magnitude of u. a)u = 20i + 25j b)u = -4j

7. Angle Between Two Vectors If θ is the angle between two nonzero vectors u and v, then → the angle you are finding must be the angle between the 2 initial points of the vectors u θ v

8. Ex 4) Find the angle θ between the vectors a)u = (-1, 0)b) u = 3i + 4j v = (0, 2)v = -2i + 3j

9. Parallel and Orthogonal Vectors The vectors u and v are parallel if there is a scalar k such that u = k v The vectors u and v are orthogonal (perpendicular) if u · v = 0

10. Ex 5) Determine whether u and v are orthogonal, parallel, or neither. a)u = (-12, 3)b) u = -2i – 2j v = (3, 12)v = i + j