Question

If a unit vector is represented by \(\overset{⃗}{U} = 0.9\,\hat{i}-0.2\,\hat{j}+m\hat{k}\), then the value of m is

• A. \(0.85\)
• B. \(\sqrt{0.15}\)
• C. \(1\)
• D. \(\sqrt{0.77}\)

Hint:

A unit vector has magnitude 1, so the squares of its components add up to 1.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
A unit vector is a vector of magnitude exactly one. The magnitude of a vector is the square root of the sum of the squares of its components.

Step 2: Key Formula or Approach:
\[ |\vec U| = \sqrt{(0.9)^2 + (-0.2)^2 + m^2} = 1 \]

Step 3: Detailed Explanation:
Square both sides:
\[ 0.81 + 0.04 + m^2 = 1 \]
\[ m^2 = 1 - 0.85 = 0.15 \]
\[ m = \sqrt{0.15} \]
Check the other options: \(m = 0.85\) would give \(m^2 = 0.7225\), and the total would be about 1.57. The choice \(m = 1\) gives total 1.85. The choice \(m = \sqrt{0.77}\) gives total 1.62. All of these exceed 1, so none of them is a unit vector.

Final Answer:
The value of \(m\) is \(\sqrt{0.15}\), option (B). \[ \boxed{\sqrt{0.15} \text{ (B)}} \]