PURE MATHEMATICS - Vectors

 

The Scalar Product

 

 

Introduction

Rules

Example #1

Example #2

 

 

 

Introduction

 

The Scalar Product (or Dot Product), of two vectors a and b is written

 

 

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 = abcosθ

 

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)      Where λ is a scalar constant.

 

 

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

 

i) What is the vector equation describing the straight line passing through the points A(-8, 1, -2) and B(10, -1, 3)?

 

ii) Find the coordinates of a point P on AB such that OP is perpendicular to AB(origin O), hence find the distance OP to 2 d.p.

 

 

scalar product problem#2

 

 

scalar product problem#2

 

 

 

 

 

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