Question

On the vector $\hat{i} + \hat{j}$ the projection of vector $\hat{i} - \hat{j}$ will be:

• A. $\dfrac{1}{\sqrt{2}}$
• B. $\sqrt{2}$
• C. $1$
• D. $0$

Hint:

If two vectors are perpendicular, their dot product is zero, and hence the projection of one on the other is zero.

Answer 1

The Correct Option is D

Step 1: Recall projection formula.
The projection of vector $\vec{a}$ on vector $\vec{b}$ is given by: \[ \text{Projection} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|} \]

Step 2: Identify vectors.
\[ \vec{a} = \hat{i} - \hat{j}, \vec{b} = \hat{i} + \hat{j} \]

Step 3: Compute dot product.
\[ \vec{a} \cdot \vec{b} = (1)(1) + (-1)(1) = 1 - 1 = 0 \]

Step 4: Apply formula.
\[ \text{Projection} = \frac{0}{|\vec{b}|} = 0 \]

Step 5: Conclusion.
The projection is $0$, hence the correct answer is (D).