Examples on Resultant Vector Formula

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

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.