The value of x for which the angle between th | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) Let $\vec{a} = x\hat{i} - 3\hat{j} - \hat{k}$ and $\vec{b} = 2x\hat{i} + x\hat{j} - \hat{k}$.Condition $1$: The angle between $\vec{a}$ and $\vec{

Vedclass · Accepted Answer

None of these

Vedclass · Answer

(D) Let $\vec{a} = x\hat{i} - 3\hat{j} - \hat{k}$ and $\vec{b} = 2x\hat{i} + x\hat{j} - \hat{k}$.Condition $1$: The angle between $\vec{a}$ and $\vec{b}$ is acute,so $\vec{a} \cdot \vec{b} > 0$.$\vec{a} \cdot \vec{b} = (x)(2x) + (-3)(x) + (-1)(-1) = 2x^2 - 3x + 1 > 0$.Solving $2x^2 - 3x + 1 > 0$,we get $(2x - 1)(x - 1) > 0$,which implies $x  1$.Condition $2$: The angle between $\vec{b}$ and the $y$-axis (unit vector $\hat{j}$) is obtuse,so $\vec{b} \cdot \hat{j} $\vec{b} \cdot \hat{j} = (2x\hat{i} + x\hat{j} - \hat{k}) \cdot (0\hat{i} + 1\hat{j} + 0\hat{k}) = x Combining both conditions: $x  1$).The intersection is $x Since none of the options match $x < 0$,the correct answer is $D$.

The value of $x$ for which the angle between the vectors $\vec{a} = x\hat{i} - 3\hat{j} - \hat{k}$ and $\vec{b} = 2x\hat{i} + x\hat{j} - \hat{k}$ is acute,and the angle between the vector $\vec{b}$ and the $y$-axis (axis of ordinate) is obtuse,are:

Similar Questions

The points represented by $\vec{a}, \vec{b}, \vec{c}, \vec{d}$ are coplanar and $(\sin A)\vec{a} + (2\sin 2B)\vec{b} + (3\sin 3C)\vec{c} - 4\vec{d} = \vec{0}$. Then the least value of $\frac{21}{8}(\sin^2 A + \sin^2 2B + \sin^2 3C)$ is:

If $a, b, c$ are distinct real numbers and $P, Q, R$ are three points whose position vectors are respectively $a \hat{i}+b \hat{j}+c \hat{k}$,$b \hat{i}+c \hat{j}+a \hat{k}$ and $c \hat{i}+a \hat{j}+b \hat{k}$,then $\angle Q P R=$

If $\bar{a}, \bar{b}, \bar{c}$ are three unit vectors such that $|\bar{a}+\bar{b}+\bar{c}|=1$ and $\bar{b}$ is perpendicular to $\bar{c}$. If $\bar{a}$ makes angles $\alpha$ and $\beta$ with $\bar{b}$ and $\bar{c}$ respectively,then the value of $\cos \alpha+\cos \beta$ is:

Find the sine of the angle between the vectors $\vec{a}=3 \hat{i}+\hat{j}+2 \hat{k}$ and $\vec{b}=2 \hat{i}-2 \hat{j}+4 \hat{k}$.

Let $ABC$ be a triangle. Let $u = \overrightarrow{AB}$ and $v = \overrightarrow{AC}$. If $D$ is the midpoint of $BC$,then the length of the median of $\triangle ABD$ through the vertex $B$ is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module