Let \(\vec{a}\) and \(\vec{b}\) be perpendicular vectors with \(|\vec{a}| = 3\) and \(|\vec{b}| = 4\). Which of the following statements are correct?
(A) \(|\vec{a}+\vec{b}| = 5\)
(B) \(|\vec{a}\times\vec{b}| = 12\)
(C) \((\vec{a}+\vec{b})\cdot(\vec{a}-\vec{b}) = 7\)
(D) The angle between \(\vec{a}+\vec{b}\) and \(\vec{a}\) is \(\cos^{-1}\left(\displaystyle\frac{3}{5}\right)\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A), (B) and (D) only
Since \(\vec{a}\) and \(\vec{b}\) are perpendicular, \(\vec{a}\cdot\vec{b} = 0\) and \(\sin\theta = 1\):
\(|\vec{a}+\vec{b}|^2 = 9+16+0 = 25\)
\(|\vec{a}\times\vec{b}| = 3\times4\times1 = 12\)
\((\vec{a}+\vec{b})\cdot(\vec{a}-\vec{b}) = |\vec{a}|^2-|\vec{b}|^2 = 9-16 = -7\)
\((\vec{a}+\vec{b})\cdot\vec{a} = |\vec{a}|^2+0 = 9\)
So \(|\vec{a}+\vec{b}| = 5\), \(|\vec{a}\times\vec{b}| = 12\), \((\vec{a}+\vec{b})\cdot(\vec{a}-\vec{b}) = -7\) and \((\vec{a}+\vec{b})\cdot\vec{a} = 9\).
Checking the statements
(A) \(|\vec{a}+\vec{b}| = 5\) — Correct. \(\sqrt{25} = 5\).
(B) \(|\vec{a}\times\vec{b}| = 12\) — Correct. \(3\times4\times\sin\displaystyle\frac{\pi}{2} = 12\).
(C) \((\vec{a}+\vec{b})\cdot(\vec{a}-\vec{b}) = 7\) — Incorrect. It is \(9-16 = -7\).
(D) The angle is \(\cos^{-1}\left(\displaystyle\frac{3}{5}\right)\) — Correct. \(\cos\theta = \displaystyle\frac{9}{5\times3} = \frac{3}{5}\).