Question

For vectors \( \vec{a}=3\hat{i}-\hat{j}+2\hat{k} \) and \( \vec{b}=\hat{i}+2\hat{j}-\hat{k} \), find the scalar projection of \( \vec{a} \) onto \( \vec{b} \).

• A. \( -\frac1{\sqrt6} \)
• B. \( \frac1{\sqrt6} \)
• C. \( -\frac16 \)
• D. \( \frac16 \)

Hint:

A negative scalar projection means the vector has a component opposite to the direction of the reference vector.

Answer 1

The Correct Option is A

Concept: The scalar projection of vector \( \vec{a} \) on vector \( \vec{b} \) is given by: \[ \text{Scalar Projection}= \frac{\vec{a}\cdot\vec{b}}{|\vec{b}|} \]

Step 1: Find the dot product:
\[ \vec{a}\cdot\vec{b} = (3)(1)+(-1)(2)+(2)(-1) \] \[ =3-2-2=-1 \]

Step 2: Find the magnitude of \( \vec{b} \):
\[ |\vec{b}| = \sqrt{1^2+2^2+(-1)^2} \] \[ = \sqrt{1+4+1} = \sqrt6 \]

Step 3: Calculate scalar projection:
\[ \frac{\vec{a}\cdot\vec{b}}{|\vec{b}|} = \frac{-1}{\sqrt6} \]