Download App

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

Gujarat BoardEnglish MediumSTD 11 ScienceJEESTD 12 - 10. vector algebra4 Marks
MCQ
Let $\vec a = 2\hat i + \hat j - 2\hat k$ and $\vec b = \hat i + \hat j$ . Let $\vec c$ be vector such that $\left| {\vec c - \vec a} \right| = 3,\;\left| {\left( {\vec a \times \vec b} \right) \times \vec c} \right| = 3$ and the angle between $\vec c$ and $\vec a \times \vec b$ be $30^\circ $ . Then $\vec a \cdot \vec c$ is equal to :
  • A
    $\frac{1}{8}$
  • B
    $\frac{{25}}{8}$
  • ✓
    $2$
  • D
    $5$
✓

Answer

Correct option: C.
$2$
c
$\overrightarrow{\mathrm{a}}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-2 \hat{\mathrm{k}}, \quad \overrightarrow{\mathrm{b}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}$

$\Rightarrow|\overrightarrow{\mathrm{a}}|=3$

$\therefore \vec{a} \times \vec{b}=2 \hat{i}-2 \hat{j}+\hat{k}$

$|\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}|=\sqrt{2^{2}+2^{2}+1^{2}}=3$

We have $|(\vec{a} \times \vec{b}) \times \vec{c}|=|\vec{a} \times \vec{b} \| \vec{c}| \sin 30^{\circ}$

$\Rightarrow|(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}) \times \overrightarrow{\mathrm{c}}|=3|\overrightarrow{\mathrm{c}}| \cdot \frac{1}{2}$

$\Rightarrow 3=3|\overrightarrow{\mathrm{c}}| \cdot \frac{1}{2}$

$\therefore|\overrightarrow{\mathrm{c}}|=2$

Now $|\overrightarrow{\mathrm{c}}-\overrightarrow{\mathrm{a}}|=3$

On squaring, we get

$\Rightarrow c^{2}+a^{2}-2 \vec{c} \cdot \vec{a}=9$

$\Rightarrow 4+9-2 \vec{a} \cdot \vec{c}=9$

$\Rightarrow \overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{c}}=2$

[$\because $ $\vec c.\vec a = \vec a.\vec c$]

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

A person forgets his $4-$digit $ATM$ pin code. But he remembers that in the code all the digits are different, the greatest digit is $7$ and the sum of the first two digits is equal to the sum of the last two digits. Then the maximum number of trials necessary to obtain the correct code is $...........$.
→
If the solution curve $y=y(x)$ of the differential equation $y ^{2} dx +\left( x ^{2}- xy + y ^{2}\right) dy =0$, which passes through the point $(1,1)$ and intersects the line $y=\sqrt{3} x$ at the point $(\alpha, \sqrt{3} \alpha)$, then value of $\log _{e}(\sqrt{3} \alpha)$ is equal to
→
Let $A$ be a $2 \times 2$ real matrix and $I$ be the identity matrix of order $2$ . If the roots of the equation $|A-x I|=0$ be $-1$ and $3$ , then the sum of the diagonal elements of the matrix $A^2$ is$..............$
→
If $P \equiv (x,\;y)$, ${F_1} \equiv (3,\;0)$, ${F_2} \equiv ( - 3,\;0)$ and $16{x^2} + 25{y^2} = 400$, then $P{F_1} + P{F_2}$ equals
→
The number of elements in the set $S =\left\{\theta \in[0,2 \pi]: 3 \cos ^4 \theta-5 \cos ^2 \theta-2 \sin ^2 \theta+2=0\right\}$ is $...........$.
→
The function $f(x) = {x^3} - 3{x^2} - 24x + 5$ is an increasing function in the interval given below
→
An arithmetic progression is written in the following way $Image$ The sum of all the terms of the $10^{\text {th }}$ row is..........
→
Let $\vec a = 3\hat i + 2\hat j + 2\hat k$ and $\vec b = \hat i + 2\hat j - 2\hat k$ be two vectors. If a vector perpendicular to both the vectors $\vec a + \vec b$ and $\vec a - \vec b$ has the magnitude $12$ then one such vector is
→
If $\alpha, \beta$ are the roots of the equation, $x^2-x-1=0$ and $S_n=2023 \alpha^n+2024 \beta^n,$ then
→
If the roots of the equations ${x^2} - bx + c = 0$ and ${x^2} - cx + b = 0$ differ by the same quantity, then $b + c$ is equal to 
→