Dot Product Of Vectors

The Dot Product of Vectors can be defined in two ways

Geometrically

Algebraically

Dot Product Of Vectors

Dot Product – Geometrical Definition

The Dot Product of Vectors is written as a.b=|a||b|cosϴ

Where |a| and |b| are the magnitudes of vector a and b and ϴ is the angle between vector a and b.

If the two vectors are Orthogonal, i.e., the angle between them is 90 then a.b=0 as cos 90 is 0.

If the two vectors are parallel to each other the a.b=|a||b| as cos 0 is 1.

Dot Product – Algebraic Definition

The Dot Product of Vectors is written as

Dot product of vector – An example

Let there be two vectors [6,2,-1] and [5,-8,2]

a.b=(6)(5)+(2)(-8)+(-1)(2)

a.b=30-16-2

a.b=12

Let there be two vectors |a|=4 and |b|=2 and ϴ=60

a.b=|a||b|cos 60

a.b=4.2cos60

a.b=4

Dot Product of Vector – Properties

The following are the Vector properties:

Commutative property

a.b=b.a

a.b=|a||b|cosϴ

a.b=|b||a|cosϴ

Distributive property

a.(b+c)=a.b+a.c

Bilinear property

a.(rb+c)=r.(a.b)+(a.c)

Scalar Multiplication property

(xa).(yb)=xy(a.b)

Non-Associative property

Because a dot product between a scalar and a vector is not allowed

Orthogonal property

Two vectors are orthogonal only if a.b=0

Dot Product of Vector – Valued Functions

The dot product of vector-valued functions, r(t) and u(t) each gives you a vector at each particular “time” t, and so the function r(t)⋅u(t) is a scalar function.

Inferring, the dot product at each given t; this product precisely measures the relationship between r(t) and u(t).


Practise This Question

Ted went to a nursery and bought some saplings. At the nursery, grass beds were being sold. In order to put them on his lawn, he bought 28 rectangular grass beds measuring 10 cm by 15 cm. When he went back and arranged them on his lawn, 4 beds were left unused. Find the area of the lawn (in square cm).