6.4 Vectors and Dot Products
The Dot Product of Two Vectors
The symbol of Dot Product is "• " and written like: u • v
and spoken like the Dot Product of vector u and vector v
Definition of a Dot Product:
The Dot Product of u = < u1, u2> and v = <v1, v2 > is:
u • v = (u1)( v1) + ( u2)(v2)
1. u • v = v • u
2. 0 • v = 0
3. u • (v + w) = u • v + u • w
4. v • v = ll v ll 2
5. c (u • v) = cu • v = u • cv
Example 1: Find the dot product
< 2, 3 > • < -1, 4 > = (2)(-1) + (3)(4) = -2 + 12 = 10
Example 2: u = < 5, 12 > and v = < -3, 2 >
u • v = (5)(-3) + (12)(2) = -15 + 24 = 9
Example 3: u = < 2, 2 > and v = < -3, 4 > and w = < 1, -4 >
a) ll u ll - 2 = √(22+2 2) - 2 = √(4 + 4 ) - 2 = √(8) - 2 = 2√2 - 2
b) w • (u + v) = w • u + w • v
= ((1)(2) + (-4)(2)) + ((1)(-3) + (-4)(4))
= (2 + -8) + (-3 + -16 ) = - 6 + -19 = -25
c) (w • u ) v = ((1)(2) + (-4)(2)) v
= (2 + -8) v
= (-6) v
= (-6) < -3, 4 >
= < 18, -24 >
d) 4u • v = 4 (u • v)
= 4 ((2)(-3) + (2)(4))
= 4 ( -6 + 8)
= 4 (2) = 8
Use the Dot Product to find ll u ll
u = < 2, -4 >
ll u ll 2 = u · u = (2)(2) + (-4)(-4) = 4 + 16
ll u ll = √20
ll u ll = √(( 2 2 +(-4)2 )= √(4 + 16) = √20
As you can see, it gives you the same answer!
The angle between two non-zero vectors is the angle θ, 0π ≤ θ ≤ π
between its respective standard position vectors. This angle can be found using
the dot product.
If θ is the angle between two nonzero vectors u and v, then
(ll u ll )(llvll )(Cos θ) = u • v
Check out this web site:
http://mathproofs.blogspot.com/2006/07/dot-product-and-cosine.html
Example : Find the angle θ between the vectors u and v where u = < 4, 4 > and v = < -2, 0 >
(ll u ll )(ll v ll )(Cos θ) = u • v
(ll< 4, 4 > ll )(ll< -2, 0 >ll )(Cos θ) = < 4, 4 > • < -2, 0 >
(√(42 + 42)) √((-2)2 + (0)2) (Cos θ) = (4)(-2) + (4)(0)
(√(16 + 16)) √(4 ) (Cos θ) = -8
√(32) √(4 ) (Cos θ) = -8
(4√2)(2) (Cos θ) = -8
Cos θ = -√(2)/2
θ = -45 degrees or -π/4 and since the angle has to be in the first or second quadrant and cosine is negative, this would be the second quadrant or
θ = 3π/4
Definition of Orthogonal Vectors
The vectors u and v are orthogonal if u • v = 0
The terms orthogonal and perpendicular mean essentially the same thing - meeting at right angles.
Example : Are the vectors u = < 3, -3 > and v = < -1, -1 > orthogonal, parallel, or neither?
u • v = (3)(-1) + (-3)(-1) = -3 + 3 = 0 so they are orthogonal
Example: u = j and v = i - 2j so u = < 0, 1 > and v = < 1, -2 >
u • v = (0)(1) + (1)(-2) = 0 + -2 = -2 so they are not orthogonal
m1 = 1/0 and m2 = -2/1 so they are not parallel so they are neither
Example 3: u = < 3, 3 > and v = < 5, 5 >
u • v = (3)(5) + (3)(5) = 15 + 15 = 30 so they are not orthogonal
m1 = 3/3 = 1 and m2 = 5/5 = 1 so they are parallel
