Question

If \(\vec{a} = 0.4\hat{i} + 0.3\hat{j} + b\hat{k}\) is a unit vector, then the value of b is

• A. \({\sqrt{3}}\)
• B. \(\dfrac{2}{\sqrt{5}}\)
• C. \(\dfrac{\sqrt{5}}{2}\)
• D. \(\dfrac{1}{\sqrt{3}}\)
• E. \(\dfrac{\sqrt{3}}{2}\)

Hint:

For unit vectors, remember that their magnitude should always be equal to 1. Use this property to solve for unknown components.

Answer 1

Answer: (E)

Step 1: The magnitude of a unit vector is 1, so: \[ |\vec{a}| = \sqrt{(0.4)^2 + (0.3)^2 + b^2} = 1 \]
Step 2: Solving for \( b \), we get: \[ b^2 = 0.75 \] \[ b = \sqrt{0.75} \] \[ b = 0.866 \] \[ b = \sqrt{\frac{3}{2}} \]
\[ \sqrt{0.16 + 0.09 + b^2} = 1 \implies b^2 = 1 - 0.25 = 0.75 \]
Step 3: Hence, \[ b = \dfrac{\sqrt{3}}{2} \] Therefore, the correct answer is (E).