Resultant Vector Formula

In mathematics, we often heard about the two terms scalar and vector. Scalar is a quantity which has only magnitude i.e. Scalar quantity describes the distance. On the other hand, Vector is a quantity which deals with both magnitude and direction. Vector quantity has both magnitude and direction.

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 with respect to others. This formula has various applications in Engineering & Physics. Based on the direction of a vector with respect 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” are 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 opposite direction with respect 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” are two vectors inclined to each other at an angle θ, then the resultant vector “r” is given by-

r = a² + b² + 2abcosθ

Here a², b² represents magnitude of the vector a, b.

Resultant vector representation

Sample Problems

Question 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

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

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

r = a + b

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

= 5i+10j+15k

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

Question 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

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

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

r = a + b

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

= 3i-6j+15k

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

Question 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

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

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

r = a – b

= (2i-2j+k) – (2i+7j+3k)

= 0i-9j-2k

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

Question 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

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

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

r = a – b

= (9i+2j-3k) – (i-3j+2k)

= 8i+5j-5k

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

Question 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

The resultant vector from the given vectors is 29.65

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

Answer:

Given,

Magnitude of vector a (a²)=2

Magnitude of vector b (b²)=4

θ = 45°

So the 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

The resultant vector from the given vectors is 17.31