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 = < u

_{1}, u

_{2}> and v = <v

_{1}, v

_{2}> is:

u • v = (u

_{1})( v

_{1}) + ( u

_{2})(v

_{2})

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 = √(2

^{2}+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 >

(√(4

^{2}+ 4

^{2})) √((-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

m

_{1}= 1/0 and m

_{2}= -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

m

_{1}= 3/3 = 1 and m

_{2}= 5/5 = 1 so they are parallel