Question

The projection of vector \(\vec{A}=\hat{i}-2\hat{j}+\hat{k}\) on vector \(\vec{B}=4\hat{i}-4\hat{j}+7\hat{k}\) is:

• A. \( \frac{17}{9} \)
• B. \( \frac{17}{7} \)
• C. \( \frac{19}{7} \)
• D. \( \frac{19}{9} \)

Hint:

Be careful to distinguish between scalar projection (which is a number, as asked for here) and vector projection (which would be a vector: \( \frac{\vec{A} \cdot \vec{B}}{|\vec{B}|^2}\vec{B} \)). The formula divides by the magnitude \(|\vec{B}|\), not the magnitude squared.

Answer 1

The Correct Option is D

Step 1: Recall the formula for the scalar projection of vector \(\vec{A}\) onto vector \(\vec{B}\). The projection is given by: \[ \text{proj}_{\vec{B}}\vec{A} = \frac{\vec{A} \cdot \vec{B}}{|\vec{B}|} \]
Step 2: Calculate the dot product \(\vec{A} \cdot \vec{B}\). \[ \vec{A} \cdot \vec{B} = (1)(4) + (-2)(-4) + (1)(7) \] \[ \vec{A} \cdot \vec{B} = 4 + 8 + 7 = 19 \]
Step 3: Calculate the magnitude of vector \(\vec{B}\). \[ |\vec{B}| = \sqrt{4^2 + (-4)^2 + 7^2} \] \[ |\vec{B}| = \sqrt{16 + 16 + 49} = \sqrt{81} = 9 \]
Step 4: Compute the projection. \[ \text{proj}_{\vec{B}}\vec{A} = \frac{19}{9} \]