If the angle between the vectors x i +3 j -7 k and x i -x j +4 k … - Vidyadip

MCQ

If the angle between the vectors $\text{x}\hat{\text{i}}+3\hat{\text{j}}-7\hat{\text{k}}$ and $\text{x}\hat{\text{i}}-\text{x}\hat{\text{j}}+4\hat{\text{k}}$ is acute, then $x$ lies in the interval:

A
$(-4, 7)$
B
$[-4, 7]$
✓
$R - [-4, 7]$
D
$R - (4, 7)$

✓

Answer

Correct option: C.

$R - [-4, 7]$

Let $\theta$ be the angle between $\vec{\text{a}}$ and $\vec{\text{b}}.$
$\cos\theta=\frac{\vec{\text{a}}.\vec{\text{b}}}{|\vec{\text{a}}|\big|\vec{\text{b}}\big|}=\frac{\text{x}^2-3\text{x}-28}{\sqrt{\text{x}^2+3^2+49}\sqrt{\text{x}^2+\text{x}^2+4^2}}$
For $\theta$ to be acute,
$\cos\theta>0$
$\Rightarrow\text{x}^2-3\text{x}-28>0$
$\Rightarrow(\text{x}-7)(\text{x}+4)>0$
$\Rightarrow\text{x}\in(-\infty,-4)\cup(7,\infty)$
$\Rightarrow\text{x}\in\text{R}-[-4,7]$

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