Download App

STD 12 - 10. vector algebra — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEESTD 12 - 10. vector algebra4 Marks
MCQ
Let $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}$ and $\overrightarrow{\mathrm{c}}$ be three unit vectors such that $|\vec{a}-\vec{b}|^{2}+|\vec{a}-\vec{c}|^{2}=8$ Then $|\vec{a}+2 \vec{b}|^{2}+|\vec{a}+2 \vec{c}|^{2}$ is equal to
  • A
    $1$
  • ✓
    $2$
  • C
    $4$
  • D
    $6$
✓

Answer

Correct option: B.
$2$
b
$|\vec{a}|=|\vec{b}|=|\vec{c}|=1$

$|\vec{a}-\vec{b}|^{2}+|\vec{a}-\vec{b}|^{2}=8$

$\Rightarrow \quad|\vec{a}|^{2}+|\vec{b}|^{2}-2 \vec{a} \cdot \vec{b}+|\vec{a}|^{2}+|\vec{c}|^{2}-2 \vec{a} \cdot \vec{c}=8$

$\Rightarrow \quad 4-2(\vec{a} \cdot \vec{b}+\vec{a} \cdot \vec{c})=8$

$\Rightarrow \quad \vec{a} \cdot \vec{b}+\vec{a} \cdot \vec{c}=-2$

$|\overrightarrow{\mathrm{a}}+2 \overrightarrow{\mathrm{b}}|^{2}+|\overrightarrow{\mathrm{a}}+2 \overrightarrow{\mathrm{c}}|^{2}$

$=\left|\mathrm{a}^{2}\right|+4|\overrightarrow{\mathrm{b}}|^{2}+4 \overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}+|\overrightarrow{\mathrm{a}}|^{2}+4|\overrightarrow{\mathrm{c}}|^{2}+4 \overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{c}}$

$=10+4(\vec{a} \cdot \vec{b}+\vec{a} \cdot \vec{c})$

$=10-8$

$=\sqrt{2}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

If $(a, b, c)$ is the image of the point $(1,2,-3)$ in the line, $\frac{x+1}{2}=\frac{y-3}{-2}=\frac{z}{-1},$ then $a+b+c$ is equal to
→
The locus of the point of intersection of the perpendicular tangents to the parabola ${x^2} = 4ay$ is
→
If $A$ and $B$ are two events such that $P\,(A) = \frac{3}{8},\,$ $P\,(B) = \frac{5}{8}$ and $P\,(A \cup B) = \frac{3}{4},$ then $P\,\left( {\frac{A}{B}} \right) = $
→
Let $X$ have a binomial distribution $B ( n , p )$ such that the sum and the product of the mean and variance of $X$ are $24$ and $128$ respectively. If $P ( X > n -3)=\frac{ k }{2^{ n }}$, then $k$ is equal to.
→
$\int_{ - 3}^3 {\frac{{{x^2}\sin 2x}}{{{x^2} + 1}}\,dx = } $
→
Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a polynomial function of degree four having extreme values at $\mathrm{x}=4$ and $\mathrm{x}=5$.
If $\lim _{x \rightarrow 0} \frac{f(\mathrm{x})}{\mathrm{x}^{2}}=5$, then $f(2)$ is equal to :
→
If ${\log _3}2,\;{\log _3}({2^x} - 5)$ and ${\log _3}\left( {{2^x} - \frac{7}{2}} \right)$ are in $A.P.$, then $x$ is equal to
→
There are two candles of same length and same size. Both of them burn at uniform rate. The first one burns in $5\,hr$ and the second one burns in $3\,h$. Both the candles are lit together. After how many minutes the length of the first candle is $3$ times that of the other?
→
A bag contains $30$ balls numbered from $1$ to $30$, one ball is drawn randomly. The probability that number on the ball is multiple of $5$ or $7$ is
→
If $f(x) = \left\{ \begin{array}{l}{2^{1/x}},{\rm{for\,\,}}\,x \ne 0\\\,\,\,\,\,\,\,3,{\rm{for\,\,}}\,x = {\rm{0}}\end{array} \right.$, then
→