Question

The projection of the vector \[ \vec{a} = \hat{i} - 2\hat{j} + \hat{k} \] on the vector \[ \vec{b} = 4\hat{i} - 4\hat{j} + 7\hat{k} \] is:

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

Hint:

To find the projection of a vector onto another, use the formula \( \text{proj}_{\vec{b}} \vec{a} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2} \, \vec{b} \), and then compute the magnitude if required.

Answer 1

The Correct Option is D

The projection of vector \( \vec{a} \) onto vector \( \vec{b} \) is given by the formula: \[ \text{proj}_{\vec{b}} \vec{a} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2} \, \vec{b} \] First, calculate the dot product \( \vec{a} \cdot \vec{b} \): \[ \vec{a} \cdot \vec{b} = (1)(4) + (-2)(-4) + (1)(7) = 4 + 8 + 7 = 19 \] Next, calculate the magnitude squared of \( \vec{b} \): \[ |\vec{b}|^2 = (4)^2 + (-4)^2 + (7)^2 = 16 + 16 + 49 = 81 \] Now, use the formula for the projection: \[ \text{proj}_{\vec{b}} \vec{a} = \frac{19}{81} \, \vec{b} \] The magnitude of the projection is: \[ \left| \text{proj}_{\vec{b}} \vec{a} \right| = \frac{19}{81} \times \sqrt{81} = \frac{19}{9} \] Thus, the magnitude of the projection is: \[ \boxed{19} \]