6.4 Vectors and Dot Products

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. Dot Product

The dot product of u = and v = is given by u●v = u 1 v 1 + u 2 v 2. Definition of Dot Product

Let u, v, and w be vectors in the plane or in space and let c be a scalar. 1.u●v = v●u 2.0●v = 0 3.u●(v + w) = u●v + u●w 4.v●v = ||v|| 2 5.c(u●v) = cu●v = u●cv Properties of the Dot Product

Find each dot product. A) ● B) ● C) ● D) ● E) (5i + j)●(3i – j) Example 1: Finding Dot Products

Let u =, v = and w =. Find the dot product. A) (u●v)w B) u●2v Example 2: Using Properties of Dot Products

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

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

Find the angle between u = and v =. Example 4: Finding the Angle Between Two Vectors

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

Are the vectors u = and v = orthogonal? Example 5: Determining Orthogonal Vectors