Let a = 2 i + j - 2 k and b = i + j . Let c be vector such that |… - Vidyadip

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

More "M.C.Q (1 Marks)" from 10. vector algebra→10. vector algebra — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

The value of c in Lagrange's mean value theorem for the function f(x) = x(x - 2) when $\text{x}\in[1,2]$ is:

$1$
$\frac{1}{2}$
$\frac{2}{3}$
$\frac{3}{2}$

If $\int_0^\pi {xf(\sin x)dx = A} \int_0^{\pi /2} {f(\sin x)dx} $, then $A$ is

Area bounded by the lines $y = x,\,\,x = - 1,\,\,x = 2$ and $x - $ axis is

The value of objective function is maximum under linear constraints

at the centre of feasible region
at (0, 0)
at any vertex of feasible region
the vertex which is maximum distance from (0, 0)

The degree of the differntial equation $\Big(\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}\Big)^{2}=\Big(\frac{\text{dy}}{\text{dx}}\Big)=\text{y}^{3}$ is:

$\frac{1}{2}$
$2$
$3$
$4$

Evaluate the following integral: $\int\sec^2\text{xdx:}$

$2\tan\text{x+c}$
$\tan2\text{x}+\text{c}$
$\tan\text{x}+\text{c}$
None of these

Which of the following is a nilpotent matrix

$\int_0^{\pi / 4} \tan ^2 x d x$ is equal to :

A function f from the set of natural numbers to the set of integers defined by $\text{f(n)}\begin{cases}\frac{\text{n}-1}{2},&\text{when n is odd}\\-\frac{\text{n}}{2},&\text{when n is even}\end{cases}$ is:

Neither one-one nor onto.
One-one but not onto.
Onto but not one-one.
One-one and onto.

If A and B are two events such that P(A) = 0.4, P(B) = 0.3 and $\text{P}(\text{A}\cup\text{B})=0.5,$ then $\text{P}(\overline{\text{B}}\cap\text{A})$ equals.

$\frac{2}{3}$
$\frac{1}{2}$
$\frac{3}{10}$
$\frac{1}{5}$