Dot Product of Vectors - Notes | Study Physics For JEE - JEE

The dot product of two different vectors that are non-zero  is denoted by a.b and is given by:
a.b = ab cos θ
wherein θ is the angle formed between a and b, and,
0 ≤ θ ≤ π
If a = 0 or b = 0, θ will not be defined, and in this case,
a.b= 0

You can define the dot product of two vectors in two different methods: geometrically and algebraically.

The geometric meaning of dot product says that the dot product between two given vectors a and b is denoted by:
a.b = |a||b| cos θ
Here, |a| and |b| are called the magnitudes of vectors a and b and θ is the angle between the vectors a and b.
If the two vectors are orthogonal, that is,  the angle between them is 90, then a.b = 0 since cos 90 = 0.
If the two vectors are parallel to each other, then a.b =|a||b| since cos 0 = 1.

The dot product algebra says that the dot product of the given two products – a = (a₁, a₂, a₃) and b= (b₁, b₂, b₃) is given by:

a.b= (a₁b₁ + a₂b₂ + a₃b₃)

Properties of Dot Product of Two Vectors 
Given below are the properties of vectors:

1. Commutative Property
   a .b = b.a
   a.b =|a| b|cos θ
   a.b =|b||a|cos θ
2. Distributive Property
   a.(b + c) = a.b + a.c
3. Bilinear Property
   a.(rb + c) = r.(a.b) + (a.c)
4. Scalar Multiplication Property
   (xa) . (yb) = xy (a.b)
5. Non-Associative Property
   Since the dot product between a scalar and a vector is not allowed.
6. Orthogonal Property
   Two vectors are orthogonal only when a.b = 0

Dot Product of Vector-Valued Functions

The dot product of vector-valued functions, that are r(t) and u(t), each gives you a vector at each particular time t, and hence, the function r(t)⋅u(t) is said to be a scalar function.

Solved Examples

Example 1: Find the dot product of a= (1, 2, 3) and b= (4, −5, 6). What kind of angle the vectors would form?

Using the formula of the dot products,
a.b = (a1b1 + a2b2 + a3b3)
You can calculate the dot product to be
= 1(4) + 2(−5) + 3(6)
= 4 − 10 + 18
= 12

Since a.b is a positive number, you can infer that the vectors would form an acute angle.

Example 2: Two vectors A and B are given by:
A = 2i − 3j + 7k and B= −4i + 2j −4k
Find the dot product of the given two vectors.

A.B = (2i − 3j +7k) . (−4i + 2j − 4k)
= 2 (−4) + (−3)2 + 7 (−4)
= −8 − 6 − 28
= −42

Key Points to Remember

• When two vectors are cross-products, the output is a vector that is orthogonal to the two provided vectors.
• The right-hand thumb rule determines the direction of the cross product of two vectors, and the magnitude is determined by the area of the parallelogram generated by the original two vectors.
• A zero vector is the cross-product of two linear vectors or parallel vectors.

Conclusion

Vector is a quantity that has both magnitude as well as direction. Few mathematical operations can be applied to vectors such as addition and multiplication. The multiplication of vectors can be done in two ways, i.e. dot product and cross product. The dot product of two vectors is the sum of the products of their corresponding components. It is the product of their magnitudes multiplied by the cosine of the angle between them. A vector's dot product with itself is the square of its magnitude.