Resultant Vector Formula: Definition, Examples

Resultant vector formula gives the resultant value of two or more vectors. The result is obtained by computing the vectors with consideration of the direction of each vector to others. This formula has various applications in Engineering & Physics.

A resultant vector is the vector that results from adding two or more vectors together. It is found by combining the magnitudes and directions of the individual vectors. The resultant vector is generally denoted using R. The magnitude ∣R∣ of the resultant vector R can be found using the Pythagorean theorem.

Based on the direction of a vector to other vectors, the Resultant Vector formula is classified into three types.

Resultant Vector 1st Formula

If the vectors are in the same direction then the resultant of the vector can be calculated by adding the vectors which are in the same direction. Let “a” and “b” be the vectors with the same direction then the resultant vector “r” is given by

r = a + b

Resultant Vector 2nd formula

If the vectors are in different directions then the resultant of the vector can be calculated by subtracting the vectors from each other. Let “b” be a vector which is in the opposite direction to vector “a” then the resultant vector “r” is given by

r = a – b

Resultant Vector 3rd Formula

If any vectors are inclined to each other at some angle then the resultant of these vectors can be calculated by this formula. Let “a”, and “b” be two vectors inclined to each other at an angle θ, then the resultant vector “r” is given by:

R = A² + B² + 2ABcosΦ

where,

A², B² represents Square of Magnitude of Vector A, B

The image added below shows vector A and B and their resultant vector R.

Articles Related to Resultant Vector Formula:

• Scalar and Vector
• Dot and Cross Products on Vectors
• Vector Operations
• Triangle Law of Vector Addition

Example 1: Find the resultant vector for the vectors i+2j+3k and 4i+8j+12k

Solution:

Given two vectors are a=i+2j+3k and b=4i+8j+12k

Direction ratios of the two vectors are in equal proportion. So two vectors are in the same direction.

Resultant vector formula for the given vectors is given by-

r = a + b

= (i+2j+3k) + (4i+8j+12k)

= 5i+10j+15k

Resultant vector from the given vectors is 5i+10j+15k

Example 2: Find the resultant vector for the vectors i-2j+5k and 2i-4j+10k

Solution:

Given two vectors are a=i-2j+5k and b=2i-4j+10k

Direction ratios of the two vectors are in equal proportion. So two vectors are in the same direction.

Resultant vector formula for the given vectors is given by-

r = a + b

= (i-2j+5k) + (2i-4j+10k)

= 3i-6j+15k

Resultant vector from the given vectors is 3i-6j+15k

Example 3: Find the resultant vector for the vectors 2i-2j+k and 2i+7j+3k

Solution:

Given two vectors are a=2i-2j+k and b=2i+7j+3k

Direction ratios of the two vectors are not in equal proportions. So two vectors are in opposite direction.

Resultant vector formula for the given vectors is given by-

r = a – b

= (2i-2j+k) – (2i+7j+3k)

= 0i-9j-2k

Resultant vector from the given vectors is 0i-9j-2k

Example 4: Find the resultant vector for the vectors 9i+2j-3k and i-3j+2k

Solution:

Given two vectors are a=9i+2j-3k and b=i-3j+2k

Direction ratios of the two vectors are not in equal proportions. So two vectors are in opposite direction.

Resultant vector formula for the given vectors is given by-

r = a – b

= (9i+2j-3k) – (i-3j+2k)

= 8i+5j-5k

Resultant vector from the given vectors is 8i+5j-5k

Example 5: Find the resultant of the vectors 2i+2j+2k and i+2j+3k which are inclined at an angle 30° to each other.

Solution:

Given two vectors are a=2i+2j+2k and b=i+2j+3k

Also given that given two vectors are inclined at an angle θ=30°

So the resultant vector formula for the given vectors is given by-

r = a² + b² + 2abcosθ

Magnitude of vector a (a²) = \sqrt{2^2+2^2+2^2}

= \sqrt{4+4+4}

=√12

a²=2√3

Magnitude of vector b (b²) = \sqrt{1^2+2^2+3^2}

= \sqrt{1+4+9}

=√14

b²=√14

r = a² + b² + 2abcosθ

= 2√3 + √14 + 2(2√3)(√14)cos30°

= 2√3 + √14 + 4(√3)(√14)(√3/2)

= 29.65

Resultant vector from the given vectors is 29.65

Example 6: Find the resultant of the vector having magnitude 2, 4 which is inclined at 45°.

Solution:

Given,

Magnitude of vector a (a²)=2

Magnitude of vector b (b²)=4

θ = 45°

So, resultant vector formula for the given vectors is given by-

r = a² + b² + 2abcosθ

= 2+4+2(2)(4)cos45°

= 6+16×(1/√2)

= 17.31

Resultant vector from the given vectors is 17.31

Q1. Resultant of Two Vectors in 2D

Given vectors:

• A = (3, 4)
• B = (1, 2)

Find the components of the resultant vector R.

Calculate the magnitude of R.

Determine the direction (angle) of R with respect to the positive x-axis.

Q2. Resultant of Two Vectors in 3D

Given vectors:

• A = (2, −1, 3)
• B = (−1, 4, 2)

Find the components of the resultant vector R.

Calculate the magnitude of R.

Q3. Resultant of Three Vectors in 2D

Given vectors:

• A = (4, 1)
• B = (−2, 3)
• C = (1, 4)

Find the components of the resultant vector R.

Calculate the magnitude of R.

Determine the direction (angle) of R with respect to the positive x-axis.

What is a Resultant Vector?

A resultant vector is the vector obtained when two or more vectors are added together. It represents the combined effect of all the vectors in both magnitude and direction.

How to Calculate a Resultant Vector?

To calculate a resultant vector, add the corresponding components of the given vectors. For vectors A and B in 2D: R = A + B = (A_x + B_x, A_y + B_y)

What is Formula for Magnitude of a Resultant Vector?

The magnitude of a resultant vector R = (Rx, Ry) is calculated using the Pythagorean theorem: ∣R∣= √(R_x²+R_y2)

​How do you Find Direction of a Resultant Vector?

The direction of a resultant vector R with components (R_x, R_y) can be found using the inverse tangent function: θ = tan⁡⁻¹(R_y/R_x)