6.4Vectors 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 = <u1, u2>

and v = <v1, v2> is given by u●v =

u1v1 + u2v2.

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●u2.0●v = 03.u●(v + w) = u●v + u●w4.v●v = ||v||2

5.c(u●v) = cu●v = u●cv

Properties of the Dot Product

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)

Example 1: Finding Dot Products

Let u = <-1, 3>, v = <2, -4> and w = <1, -2>. 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 = <4, 3> and v = <3, 5>.

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 = <2, -3> and v = <6, 4> orthogonal?

Example 5: Determining Orthogonal Vectors