If \(\vec{a}=\hat{i}+\hat{j}+\hat{k}\), \(\vec{b}=2\hat{i}-\hat{j}+3\hat{k}\) and \(\vec{c}=\hat{i}-2\hat{j}+\hat{k}\), and a unit vector parallel to the vector \(2\vec{a}-\vec{b}+3\vec{c}\) ?

Question

Question

This question was previously asked in

AAI ATC Junior Executive 25 March 2021 Official Paper (Shift 3)

\(\frac{9}{{\sqrt {22} }}\hat{i}-\frac{3}{{\sqrt {22} }}\hat{j}+\frac{1}{{\sqrt {22} }}\hat{k}\)
\(\frac{2}{{\sqrt {22} }}\hat{i}-\frac{3}{{\sqrt {22} }}\hat{j}+\frac{2}{{\sqrt {22} }}\hat{k}\)
\(\frac{3}{{\sqrt {22} }}\hat{i}-\frac{3}{{\sqrt {22} }}\hat{j}+\frac{3}{{\sqrt {22} }}\hat{k}\)
\(\frac{3}{{\sqrt {22} }}\hat{i}-\frac{3}{{\sqrt {22} }}\hat{j}+\frac{2}{{\sqrt {22} }}\hat{k}\)

Answer 1

Given:

Vector a = î + ĵ + k̂

Vector b = 2î - ĵ + 3k̂

Vector c = î - 2ĵ + k̂

Calculation:

The unit vector in the direction of the vector, r = 2a - b + 3c

⇒ r = 2(î + ĵ + k̂) - (2î - ĵ + 3k̂) + 3(î - 2ĵ + k̂)

⇒ r = 2î + 2ĵ + 2k̂ - 2î + ĵ - 3k̂ + 3î - 6ĵ + 3k̂

⇒ r = (2 - 2 + 3)î + (2 + 1 - 6)ĵ + (2 - 3 + 3)k̂

⇒ r = 3î - 3ĵ + 2k̂

Magnitude of r = √(3² + (-3)² + 2²)

⇒ |r| = √(9 + 9 + 4) = √22

Unit vector in the direction of r = (1 / |r|) × r

⇒ Unit vector = (1 / √22) × (3î - 3ĵ + 2k̂)

⇒ Unit vector = (3 / √22)î - (3 / √22)ĵ + (2 / √22)k̂

∴ The required vector is (3 / √22)î - (3 / √22)ĵ + (2 / √22)k̂