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
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