Question

If \( \mathbf{a} = 4\hat{i} + 5\hat{j} - 3\hat{k} \) and \( \mathbf{b} = 6\hat{i} - 2\hat{j} - 2\hat{k} \) are two vectors, then the magnitude of the component of \( \mathbf{b} \) parallel to \( \mathbf{a} \) is:

• A. \( 2\sqrt{2} \)
• B. \( 10\sqrt{2} \)
• C. \( 4\sqrt{2} \)
• D. \( 6\sqrt{2} \) \bigskip

Hint:

To find the projection of one vector onto another, use the dot product and divide by the magnitude of the vector onto which the projection is made.

Answer 1

The Correct Option is A

The magnitude of the projection of vector \( \mathbf{b} \) onto vector \( \mathbf{a} \) is determined using the formula: \[ \left| \text{Projection of } \mathbf{b} \text{ onto } \mathbf{a} \right| = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}|}. \] 

 Step 1: Calculate the dot product \( \mathbf{a} \cdot \mathbf{b} \). \[ \mathbf{a} \cdot \mathbf{b} = (4\hat{i} + 5\hat{j} - 3\hat{k}) \cdot (6\hat{i} - 2\hat{j} - 2\hat{k}). \] Applying the dot product formula: \[ \mathbf{a} \cdot \mathbf{b} = 4(6) + 5(-2) + (-3)(-2) = 24 - 10 + 6 = 20. \] Thus, \( \mathbf{a} \cdot \mathbf{b} = 20 \). 

Step 2: Determine the magnitude of \( \mathbf{a} \). \[ |\mathbf{a}| = \sqrt{4^2 + 5^2 + (-3)^2} = \sqrt{16 + 25 + 9} = \sqrt{50} = 5\sqrt{2}. \] \

Step 3: Compute the magnitude of the projection. \[ \left| \text{Projection of } \mathbf{b} \text{ onto } \mathbf{a} \right| = \frac{20}{5\sqrt{2}} = \frac{4}{\sqrt{2}} = 2\sqrt{2}. \] Hence, the magnitude of the projection of \( \mathbf{b} \) onto \( \mathbf{a} \) is: \[ \boxed{2\sqrt{2}}. \]