Question:
If \(\bar{a}=\hat{i}+2\hat{j}+2\hat{k}\), \(\bar{b}=2\hat{i}-2\hat{j}+\hat{k}\) and \(\bar{c}\) is a vector projection of \(\bar{a}\) on the line of \(\bar{b}\), then the value of \([2\bar{a}\ 3\bar{b}\ \sqrt{3}\bar{c}]\) is
If \(\bar{a}=\hat{i}+2\hat{j}+2\hat{k}\), \(\bar{b}=2\hat{i}-2\hat{j}+\hat{k}\) and \(\bar{c}\) is a vector projection of \(\bar{a}\) on the line of \(\bar{b}\), then the value of \([2\bar{a}\ 3\bar{b}\ \sqrt{3}\bar{c}]\) is
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The projection of a on b is a vector along b, so the three vectors in the triple product are linearly dependent.
Updated On: Nov 12, 2026
- \(6\sqrt{3}\)
- \(0\)
- \(\sqrt{3}\)
- \(1\)
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The Correct Option is B
Solution and Explanation
Step 1: Understanding the Concept
The vector projection of \(\bar{a}\) on \(\bar{b}\) is \(\bar{c}=\frac{\bar{a}\cdot \bar{b}}{|\bar{b}|^{2}}\bar{b}\), a scalar multiple of \(\bar{b}\).
Step 2: Compute a . b
\(\bar{a}\cdot \bar{b}=2-4+2=0\), so \(\bar{c}=\bar{0}\).
Step 3: Evaluate the triple product
A scalar triple product with a zero vector, or with two parallel vectors, is 0.
Final Answer:
The triple product is 0. \[ \boxed{\text{(B) }0} \]
The vector projection of \(\bar{a}\) on \(\bar{b}\) is \(\bar{c}=\frac{\bar{a}\cdot \bar{b}}{|\bar{b}|^{2}}\bar{b}\), a scalar multiple of \(\bar{b}\).
Step 2: Compute a . b
\(\bar{a}\cdot \bar{b}=2-4+2=0\), so \(\bar{c}=\bar{0}\).
Step 3: Evaluate the triple product
A scalar triple product with a zero vector, or with two parallel vectors, is 0.
Final Answer:
The triple product is 0. \[ \boxed{\text{(B) }0} \]
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