Question

The length of the projection of \( \mathbf{a} = 2\hat{i} + 3\hat{j} + \hat{k} \) \(\text{ on }\) \( \mathbf{b} = -2\hat{i} + \hat{j} + 2\hat{k} \) \(\text{ is equal to:}\)

• A. \( \frac{2}{3} \)
• B. \( -\frac{1}{3} \)
• C. \( -\frac{2}{3} \)
• D. \( \frac{1}{3} \)

Hint:

When finding the projection length of a vector, use the dot product formula and divide by the magnitude of the vector onto which the projection is made.

Answer 1

The Correct Option is D

We are given vectors \( \mathbf{a} = 2\hat{i} + 3\hat{j} + \hat{k} \) and \( \mathbf{b} = -2\hat{i} + \hat{j} + 2\hat{k} \). We need to find the length of the projection of \( \mathbf{a} \) onto \( \mathbf{b} \).

Step 1: Use the formula for the length of the projection.
The length of the projection of vector \( \mathbf{a} \) onto vector \( \mathbf{b} \) is given by: \[ \text{Projection length} = \frac{|\mathbf{a} \cdot \mathbf{b}|}{|\mathbf{b}|} \]

Step 2: Calculate the dot product \( \mathbf{a} \cdot \mathbf{b} \).
The dot product is calculated as: \[ \mathbf{a} \cdot \mathbf{b} = (2)(-2) + (3)(1) + (1)(2) = -4 + 3 + 2 = 1 \]

Step 3: Calculate the magnitude of \( \mathbf{b} \).
The magnitude of \( \mathbf{b} \) is: \[ |\mathbf{b}| = \sqrt{(-2)^2 + 1^2 + 2^2} = \sqrt{4 + 1 + 4} = \sqrt{9} = 3 \]

Step 4: Calculate the projection length.
Now we substitute the values into the formula: \[ \text{Projection length} = \frac{|1|}{3} = \frac{1}{3} \] Thus, the correct answer is \( \frac{1}{3} \), corresponding to option (d).