12.4 The Dot Product
Given a = (a1, a2, a3) and b = (b1, b2, b3), a i b = (a1)(b1) + (a2)(b2) + (a3)(b3)
Note that this gives an answer that is NOT a vector. The dot product gives an answer that is a scalar.
Examples:
1) a = (1, -2, 3) b = (2, 0, -4)
2) a = i + 2j + k
b = 3i - 2j - k
ai b =
ai b =
Properties of the dot product:
a i a= a
2
a i b=b i a
a i 0=0 i a = 0
a i (b+c)=a i b+a i c
(b+c) i a=b i a+c i a
Law of cosines:
a
c 2 = a 2 + b2 − 2ab cos θ
c
θ
b
Using vectors:
This leaves us with
So we can say:
a i b= a b cosθ
aib
=cosθ
a b
Example:
Find the angle between a = i +2j + k and b = 3i – 2j – k.
( )
Projection of a on b a|| : projba = λub where λ is the component of a in the direction of b and
compba = a i ub
Notice that a = a|| + a ⊥ and that a|| || b and a ⊥ ⊥ b
Example: Find the projection of a on b:
a = j + 3k b = 2i – j + 2k
The angles α , β , γ that a vector makes with unit vectors i, j, and k are called direction angles of a.
Examples:
1) Find a unit vector with direction angles:
2) Find the direction angles of a = i -
3k
π 2π π
6
,
3
,
2
12.5 Cross Product
If vectors a and b are NOT parallel, they form two sides of a parallelogram:
a
b
a × b is the vector perpendicular to this plane.
So, how do we find a × b?
First we need to understand determinants:
a1 a 2
= a1 b2 − a 2 b1
b1 b2
−1 3
=
4 2
Example:
And for a 3x3:
a
d
b
e
c
f
g
h
i
=
a b
c
Show that the answer above is the same as d
e
f =
g
h
i
e
f
h
i
a−
d
f
g
i
b+
d
e
g
h
c
Formula for Cross Product:
for a = ( a1 , a2 , a3 ) and b = (b1 , b2 , b3 ) , a × b =
Example:
Calculate (3i -2j + k) × (j – 2k)
The area of this parallelogram = || a × b || = ||a|| ||b|| sin θ where θ is the angle formed by a and b.
Also, a, b, and a × b form a right hand triple:
(read page 733 in text – properties of right handed triples)
Now, if a || b then a × b = 0
Properties of Cross Product:
Anticommutative:
Scalar Factorization:
Distributive Laws:
Some identities:
a × (b × c) = (a i c)b - (a i b)c
(a × b) × c = (c i a)b - (c i b)a
(a × b) i (c × d) = (a i c)(b i d) − (a i d)(b i c)
Scalar triple product: (a × b) i c
a, b, c is a right handed triple IFF (a × b) i c > 0
Parallelepiped:
Volume =
Example: Find a vector N that is perpendicular to the plane determined by points P, Q and R and
find the area of triangle PQR.
P(0,1,0)
Q(1,-1,2)
R(2,1,1)
Example: Find the volume of the parallelepiped with edges: i – 2j +k , j + k, 2i – j - k