Let us consider vector \( \vec{a}=(c\log_{2}x)\hat{\imath}-6\hat{\jmath}+3\hat{k} \) and \( \vec{b}=(\log_{2}x)\hat{\imath}+2\hat{\jmath}+(2c\log_{2}x)\hat{k} \), then which of following statements are TRUE?
(Where \( x\in\mathrm{R} \) (set of real numbers))
(A) Vector \( \vec{a} \) and \( \vec{b} \) make obtuse angle for any \( x\in(0,\infty) \) if \( c\in(-1,0) \)
(B) Vector \( \vec{a} \) and \( \vec{b} \) makes obtuse angle for any \( x\in\mathrm{R} \) if \( c\gt0 \)
(C) Vector \( \vec{a} \) and \( \vec{b} \) makes obtuse angle for any \( x\in(0,\infty) \) if \( c\in\left(-\displaystyle\frac{4}{3},-1\right] \)
(D) Vectors \( \vec{a} \) and \( \vec{b} \) makes obtuse angle for \( x\in\mathrm{R} \) if \( c\in(1,2) \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A) and (C) only
Let \( t=\log_{2}x \). Since \( \log_{2}x \) is defined only for x > 0, the vectors exist only for \( x\in(0,\infty) \), and there t takes every real value.
\( \vec{a}\cdot\vec{b}=ct^{2}-12+6ct \)
\( =c(t+3)^{2}-9c-12 \)
The angle is obtuse when \( \vec{a}\cdot\vec{b}\lt0 \) (the vectors point in exactly opposite directions only when c = −3, which none of the statements allows).
For c > 0, \( \vec{a}\cdot\vec{b} \) grows without limit as t grows, so it is positive for large x.
For c < 0, the greatest value of \( \vec{a}\cdot\vec{b} \) is \( -9c-12 \), at t = −3. It is negative when
\( -9c-12\lt0 \)
\( \displaystyle c\gt-\frac{4}{3} \)
So the angle is obtuse for every x > 0 whenever \( -\frac{4}{3}\lt c\lt0 \), while for any c > 0 some values of x give an acute angle.
Checking the statements
(A) Vector \( \vec{a} \) and \( \vec{b} \) make obtuse angle for any \( x\in(0,\infty) \) if \( c\in(-1,0) \) — Correct. \( (-1,0) \) lies inside \( \left(-\frac{4}{3},0\right) \).
(B) Vector \( \vec{a} \) and \( \vec{b} \) makes obtuse angle for any \( x\in\mathrm{R} \) if \( c\gt0 \) — Incorrect. For c > 0, \( \vec{a}\cdot\vec{b} \) is positive for large x.
(C) Vector \( \vec{a} \) and \( \vec{b} \) makes obtuse angle for any \( x\in(0,\infty) \) if \( c\in\left(-\frac{4}{3},-1\right] \) — Correct. \( \left(-\frac{4}{3},-1\right] \) lies inside \( \left(-\frac{4}{3},0\right) \).
(D) Vectors \( \vec{a} \) and \( \vec{b} \) makes obtuse angle for \( x\in\mathrm{R} \) if \( c\in(1,2) \) — Incorrect. Here c > 0, so \( \vec{a}\cdot\vec{b} \) is positive for large x.