Introduction
The Scalar product (or Dot Product), of two vectors a and b is written
a.b
If the two vectors are inclined to each other by an angle(say θ ) then the product is written
a.b = |a|.|b|cosθ or a.b = abcos θ
Even though the left hand side of the equation is written in terms of vectors, the answer is a scalar quantity.
Rules
a.b = abcos θ = b.a
when a & b are parallel, θ = 0, cos θ = 1 , a.b = ab
(unit vectors i.i = j.j = k.k = 1)
when a & b are at 90o , θ = 90o, cos θ = 0 , a.b = 0
(unit vectors: i.j = j.i = 0 j.k = k.j = 0 k.i = i.k = 0)
if a = a1i + a2j + a3k and b = b1i + b2j + b3k
then
a.b = a1b1 + a2b2 + a3b3
|a|2 = a.a = a12 + a22 + a32
a.(b + c) = a.b + a.c a.(b - c) = a.b - a.c
(a + b).c = a.c + b.c (a - b).c = a.c - b.c
(λa).b = λ(a.b) = a.(λb)
Example #1
Given that a = 3i - j + 2k and b = 2i + j - 2k ,
find a.b and the included angle between the vectors to 1 d.p.
Example #2
What is the vector equation describing the straight line passing through the points A(-8, 1, -2) and B(10, -1, 3)?
Find the coordinates of a point P on AB such that OP is perpendicular to AB(origin O), hence find the distance OP to 2d.p.
