A unit vector perpendicular to vector c and coplanar with vectors… - Vidyadip

MCQ

A unit vector perpendicular to vector $ c $ and coplanar with vectors $a$ and $b$ is

A
$\frac{{a \times (b \times c)}}{{|a \times (b \times c)|}}$
B
$\frac{{b \times (c \times a)}}{{|b \times (c \times a)|}}$
✓
$\frac{{c \times (a \times b)}}{{|c \times (a \times b)|}}$
D
None of these

✓

Answer

Correct option: C.

$\frac{{c \times (a \times b)}}{{|c \times (a \times b)|}}$

c
(c) Any vector $(r)$ in plane of $a,\,b$ must be in form of linear combination of $a$ and $b$

$\overrightarrow r = xa + yb$

Such combination is possible in alternate $(c).$

As $c \times (a \times b) = (c\,.\,b)a - (c\,.\,a)b$ …..$(i)$

Also $(i)$ is perpendicular to $c$

As $c.\,\{ (c\,.\,b)\,a - (c\,.\,a)\,b\} $$ = (c\,.\,a)(c\,.\,b) - (c\,.\,b)(c\,.\,a) = 0$

Thus unit vector perpendicular to $c$ and coplanar with $a,\,b$ is,$\frac{{c \times (a \times b)}}{{|c \times (a \times b)|}}$

Other similar concets :

$(1)$ Unit vector perpendicular to $a$ and coplanar with $b$ and $c$ is $r = \frac{{a \times (b \times c)}}{{|a \times (b \times c)|}}$.

$(2)$ Unit vector perpendicular to $b$ and coplanar with $c$ and $a$ is$r = \frac{{b \times (c \times a)}}{{|b \times (c \times a)|}}$ .

$(b)$ We know ${(a \times b)^2} + {(a\,.\,b)^2} = {a^2}{b^2} = (a\,.\,a)(b\,.\,b)$.

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

Let $R$ be a relation on the set $N$ be defined by $\{(x, y): x, y \in N, 2 x+y=41\}$. Then, $R$ is

Which of the following is correct?

Let $\vec{a}=2 \hat{i}-\hat{j}+5 \hat{k}$ and $\vec{b}=\alpha \hat{i}+\beta \hat{j}+2 \hat{k}$. If $((\vec{a} \times \vec{b}) \times \hat{i}) \cdot \hat{k}=\frac{23}{2}$, then $|\vec{b} \times 2 \hat{j}|$ is equal to.

If $f(x) = x + {1 \over x},$ $x > 0$ , then its greatest value is

Consider the system of equations $\mathrm{ax}+\mathrm{by}=0, \mathrm{cx}+\mathrm{dy}=0$, where $\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d} \in\{0,1\}$.

$STATEMENT -1$ : The probability that the system of equations has a unique solution is $3 / 8$. and $STATEMENT - 2$: The probability that the system of equations has a solution is $1$ .

If ${e^y} + xy = e$, the ordered pair $\left( {\frac{{dy}}{{dx}},\frac{{{d^2}y}}{{d{x^2}}}} \right)$ at $x = 0$ is equal to

The general solution of the differential equation $\log \left( {\frac{{dy}}{{dx}}} \right) = x + y$ is

Area bounded by the ellipse $\frac{x^2}{4}+\frac{y^2}{9}=1$ is

If $\text{f}(\text{x})=\Big(\frac{\text{x}^\text{l}}{\text{x}^\text{m}}\Big)^{\text{l}+\text{m}}\Big(\frac{\text{x}^\text{m}}{\text{x}^\text{n}}\Big)^{\text{m}+\text{n}}\Big(\frac{\text{x}^\text{n}}{\text{x}^\text{l}}\Big)^{\text{n}+1},$ the f'(x) is equal to:

1
0
x^l+m+n
None of these.

If the shortest distance between the line $\vec{r}=(-\hat{i}+3 \hat{k})+\lambda(\hat{i}-a \hat{j})$ and $\vec{r}=(-\hat{j}+2 \hat{k})+\mu(\hat{i}-\hat{j}+\hat{k})$ is $\sqrt{\frac{2}{3}}$, then the integral value of $a$ is equal to