Vectors and Dot Products

1. Vectors and Dot Products 6.4

2. Dot Product Dot Product is a third vector operation. This vector operation yields a scalar (a single number) not another vector. The dot product can be positive, zero or negative.

3. Definition of Dot Product The dot product of u = <u1, u2> and v = <v1, v2> is given by u●v = u1v1 + u2v2.

4. Properties of the Dot Product Let u, v, and w be vectors in the plane or in space and let c be a scalar. • u●v = v●u • 0●v = 0 • u●(v + w) = u●v + u●w • v●v = ||v||2 • c(u●v) = cu●v = u●cv

5. Example 1: Finding Dot Products Find each dot product. A) <4, 5>●<2, 3> B) <2, -1>●<1, 2> C) <0, 3>●<4, -2> D) <6, 3>●<2, -4> E) (5i + j)●(3i – j)

6. Example 2: Using Properties of Dot Products Let u = <-1, 3>, v = <2, -4> and w = <1, -2>. Find the dot product. A) (u●v)w B) u●2v

7. Example 3: Dot Product & Magnitude The dot product of u with itself is 5. What is the magnitude of u?

8. Angle Between Two Vectors If θ is the angle between two nonzero vectors u and v, then cosθ = u●v (u●v = ||u|| ||v||cosθ) ||u|| ||v||

9. Example 4: Finding the Angle Between Two Vectors Find the angle between u = <4, 3> and v = <3, 5>.

10. Definition of Orthogonal Vectors The vectors u and v are orthogonal if u●v = 0. Orthogonal = Perpendicular = Meeting at 90°

11. Example 5: Determining Orthogonal Vectors Are the vectors u = <2, -3> and v = <6, 4> orthogonal?