,
Vectors : Scalar Product
 

[ introduction ][ rules ][ example#1 ][ example#2 ]

 

 

 

 

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

a).b = λ(a.b) = a.(λb)

 

scalar product with angle

 

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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.

scalar product problem#1

 

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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.

scalar product problem#2

scalar product problem#2

 

 

 

 

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