Dot Product Formula with Solved Examples

Dot Product Formula

When the scalar product of two vectors is taken, we call it the dot product. It’s a scalar number calculated from the vector components after a certain operation. Only pairs of vectors with the same dimensions can use the dot product.

Main Content: 

The dot product notion states that two vectors can be multiplied to produce a scalar quantity. It is used to obtain the item. It produces two or more vector products in two or more dimensions.

According to the geometric definition of the dot product, the dot product of two vectors a and b is:

a.b = |a||b|cosθ 

Where the angle formed by two vectors, a and b, is θ. However, this formula is useful for understanding the dot product’s features. It will be easier to compute the dot product between two provided vectors if there is a formula for the dot product in terms of the vector components.

Formula:

• The dot product between standard unit vectors, i, j, and k of length one and parallel to the coordinate axes, can be seen as a first step.

In three dimensions, the standard unit vectors. In three dimensions, the standard unit vectors i, j, and k are length one vectors that point parallel to the x, y, and z axes, respectively. Therefore, we may instantly conclude that the dot product between two separate standard unit vectors is zero since the standard unit vectors are orthogonal:

i.j = i.k = j.k=0

• It is possible to compute the dot product of a unit vector and itself. The angle is zero in this situation, and cos=1 equals zero. Because the vectors are all one-dimensional, the dot products are

i.i = j.j = k.k = 1

Solved Examples

1. Calculate the dot product of a= (1, 2, 3) and b= (4, 5, 6) by multiplying them together. What kind of angle will the vectors form?

To find the dot product of three-dimensional vectors, use the formula below.

a.b = a1b1 + a2b2 + a3b3

Thus the calculation of dot product:l

a.b = a1b1 + a2b2 + a3b3

1(4)+2(-5)+3(6)

= 4 – 10 + 18 = 12

We can infer that the vectors will create an acute angle because ab is positive.

1. Calculate a = (5,9) and b = (2,4) as a dot product. What kind of angle will the vectors form?

For the dot product of two-dimensional vectors, use the formula below.

      a.b=a1b1 + a2b2 

The dot product is calculated to be

a.b = a1b1 + a2b2 

= 5(2) – 9(4)

= 10 – 36

= – 26

We can deduce that the vectors form an obtuse angle because ab is negative.