Dot Product Of Vectors

The Dot Product of Vectors can be defined in two ways



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]




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

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



Dot Product of Vector – Properties

The following are the Vector properties:

Commutative property




Distributive property


Bilinear property


Scalar Multiplication property


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

If there are 45 birds out of 150 in a zoo which are migratory. 35 of the total birds are of  Indian origin. Which kind of birds are more in number in the zoo?