If \(\vec{a} = 2\hat{\imath}-3\hat{\jmath}+\hat{k}\) and \(\vec{b} = 2\hat{\imath}+\hat{\jmath}-\hat{k}\), then which of the following statements is/are correct?
(A) \(\vec{a}\) and \(\vec{b}\) are collinear
(B) \(\vec{a}\) and \(\vec{b}\) are perpendicular
(C) Angle between \(\vec{a}\) and \(\vec{b}\) is \(\displaystyle\frac{\pi}{4}\)
(D) \(|\vec{a} +\vec{b}| = 2\sqrt{5}\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (B) and (D) only
The dot product of the two vectors is
\( \displaystyle \vec{a}\cdot\vec{b} = (2)(2) + (-3)(1) + (1)(-1) \)
\( \displaystyle = 4 - 3 - 1 = 0 \)
So \(\vec{a}\cdot\vec{b} = 0\), with both vectors non-zero.
Checking the statements
(A) \(\vec{a}\) and \(\vec{b}\) are collinear — Incorrect. The component ratios \(\frac{2}{2}, \frac{-3}{1}, \frac{1}{-1}\) are not equal.
(B) \(\vec{a}\) and \(\vec{b}\) are perpendicular — Correct. \(\vec{a}\cdot\vec{b} = 0\).
(C) Angle between \(\vec{a}\) and \(\vec{b}\) is \(\frac{\pi}{4}\) — Incorrect. Since \(\vec{a}\cdot\vec{b} = 0\), the angle is \(\frac{\pi}{2}\).
(D) \(|\vec{a} +\vec{b}| = 2\sqrt{5}\) — Correct. Adding the vectors:
\(\vec{a} + \vec{b} = 4\hat{\imath} - 2\hat{\jmath} + 0\hat{k}\)
\(|\vec{a} + \vec{b}| = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5}\)