Question

The magnitude of the projection of the vector \( -\hat{i} + 2\hat{j} - \hat{k} \) on the \(z\)-axis is

• A. 2
• B. \( \frac{1}{\sqrt{6}} \)
• C. 1
• D. \( -\frac{1}{\sqrt{6}} \)

Hint:

Projection on coordinate axes equals the corresponding component; magnitude is absolute value of that component.

Answer 1

The Correct Option is C

Step 1: Write the given vector.
\[ \vec{a} = -\hat{i} + 2\hat{j} - \hat{k}. \]

Step 2: Identify direction of \(z\)-axis.
Unit vector along \(z\)-axis is:
\[ \hat{k}. \]

Step 3: Use projection formula.
Projection of \(\vec{a}\) on \(\hat{k}\) is:
\[ \text{Projection} = \vec{a} \cdot \hat{k}. \]

Step 4: Compute dot product.
\[ \vec{a} \cdot \hat{k} = (-1)(0) + (2)(0) + (-1)(1). \]
\[ = -1. \]

Step 5: Take magnitude.
Magnitude of projection is:
\[ |-1| = 1. \]

Step 6: Interpret result.
Only the \(k\)-component contributes to projection on \(z\)-axis.

Step 7: Final conclusion.
\[ \boxed{1} \]