Dot Product

Geometric Definition of Dot Product

Let’s know the geometric definition of a dot product:

• The scalar product of two vectors is known as the dot product.

• The dot product is a scalar number obtained by performing a specific operation on the vector components.

• The dot product is only for pairs of vectors having the same number of dimensions.

• The symbol that is used for representing the dot product is a heavy dot.

• This dot product is extensively in Physics as well as in Mathematics.

Here we are going to know about dot product distributive, the geometric meaning of dot product, geometric definition of dot product, properties of scalar and vector product, dot product commutative proof, distributive law of dot product, properties of dot product of two vectors, scalar product associative, vector product is not commutative, dot product algebraic definition, dot product geometrical definition, dot product of vector properties.

Concept of Dot Product

The concept of dot product states that any two vectors can be multiplied for getting the scalar quantity. It is used for getting the product. It is giving the products of two vectors or more vectors in two dimensions or more dimensions.

The geometric definition of the dot product says that the dot product between two vectors a and b is given as: a⋅b = |a||b|cos θ, where θ is the angle between two vectors a and b. In Mathematics, this formula is generally used for understanding the properties of the dot product. A formula for the dot product in terms of the vector components will make it easier to calculate the dot product between any two given vectors.

Dot Product - Geometrical Definition

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

Where |a|, |b| are said to be the magnitudes of vector a and b and θ is the angle between vector a and b.

If any two given vectors are said to be 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| equals 4 and |b| equals 2 and θ equals 60

a.b equals |a||b|cos 60

a.b equals 4.2 cos60

a.b equals 4.

Properties of Dot Product

(au + bv) · w equals (au) · w + (bv) · w, where a and b are known to be scalars

Below is the list of properties of the dot product:

1. u · v equals |u||v| cos θ

2. u · v  equals v · u

3. u · v equals 0 when u and v are orthogonal.

4. 0 · 0 equals 0

5. |v|2 equals v · v

6. a (u·v) equals (a u) · v

7. (au + bv) · w equals (au) · w + (bv) · w

The Formula for Dot Product

1. As the first step, we may see that the dot product between standard unit vectors, that is, the vectors i, j, and k of length one, and they are parallel to the coordinate axes.

2. The standard unit vectors in 3 dimensions. The standard unit vectors in 3 dimensions, i, j, and k are length one vectors that point parallel to the x-axis, y-axis, and z-axis respectively. Since the standard unit vectors are said to be orthogonal, we can immediately conclude that the dot product between a pair of distinct standard unit vectors is always zero: i⋅j equals i⋅k equals j⋅k equals 0.

3. The dot product between a unit vector and itself can be easily computed. In this case, the angle is zero, and cos θ = 1 as θ = 0. Given that the vectors are all of length one, the dot products are i⋅i =  j⋅j = k⋅k equals to 1.

4. Since we know the dot product of unit vectors, we can simplify the dot product formula to, a⋅b = a₁b₁ + a₂b₂ + a₃b₃.

Solved Examples

Question 1) Calculate the dot product of a = (-4,-9) and b = (-1,2).

Solution: Using the following formula for the dot product of two-dimensional vectors, a⋅b = a₁b₁ + a₂b₂ + a₃b₃.

We calculate the dot product to be

= -4(-1) - 9(2)

= 4 - 18

= -14.

Question 2) Calculate the dot product of a = (-2,-4) and b = (-1,2).

Solution: Using the following formula for the dot product of two-dimensional vectors, a⋅b = a₁b₁ + a₂b₂ + a₃b₃

We calculate the dot product to be

= -2(-1) - 4(2)

= 2 - 8

= -6.