What is the length of projection of the vector \hat{i}+2\hat{j}+3\hat{k} on the vector 2\hat{i}+3\hat{j}-2\hat{k}?
- A. \frac{1}{\sqrt{17}}
- B. \frac{2}{\sqrt{17}} ✓
- C. \frac{3}{\sqrt{17}}
- D. \frac{2}{\sqrt{14}}
Correct Answer: B. \frac{2}{\sqrt{17}}
Explanation
The projection of vector \vec{a} on \vec{b} is given by \frac{\vec{a}\cdot\vec{b}}{|\vec{b}|}. Here, \vec{a}\cdot\vec{b} = (1)(2) + (2)(3) + (3)(-2) = 2 + 6 - 6 = 2. The magnitude |\vec{b}| = \sqrt{2^2 + 3^2 + (-2)^2} = \sqrt{4+9+4} = \sqrt{17}. Thus, the projection length is \frac{2}{\sqrt{17}}.
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