Let a=a_ 1 i+a_ 2 j+a_ 3 k a_ i >0, i=1,2,3 be a vector which mak… - Vidyadip

MCQ

Let $\vec{a}=a_{1} \hat{i}+a_{2} \hat{j}+a_{3} \hat{k} \quad a_{i}>0, i=1,2,3$ be $a$ vector which makes equal angles with the coordinates axes OX, OY and OZ. Also, let the projection of $\vec{a}$ on the vector $3 \hat{i}+4 \hat{j}$ be $7$ . Let $\vec{b}$ be a vector obtained by rotating $\vec{a}$ with $90^{\circ}$. If $\vec{a}, \vec{b}$ and $x$-axis are coplanar, then projection of a vector $\vec{b}$ on $3 \hat{i}+4 \hat{j}$ is equal to

A
$\sqrt{7}$
✓
$\sqrt{2}$
C
$2$
D
$7$

✓

Answer

Correct option: B.

$\sqrt{2}$

b
$\overrightarrow{ a }=a_{1} \hat{ i }+ a _{2} \hat{ j }+ a _{3} \hat{ k }$

$\overrightarrow{ a }=\lambda\left(\frac{1}{\sqrt{3}} \hat{ i }+\frac{1}{\sqrt{3}} \hat{ j }+\frac{1}{\sqrt{3}} \hat{ k }\right)=\frac{\lambda}{\sqrt{3}}(\hat{ i }+\hat{ j }+\hat{ k }$

Now projection of $\vec{a}$ on $\vec{b}=7$

$\Rightarrow \frac{\overrightarrow{ a } \cdot \overrightarrow{ b }}{|\overrightarrow{ b }|}=7$

$\frac{\lambda}{\sqrt{3}} \frac{(\hat{ i }+\hat{ j }+\hat{ k }) \cdot(3 \hat{ i }+4 \hat{ j })}{5}=7$

$\lambda=5 \sqrt{3}$

$\overrightarrow{ a }=5(\hat{ i }+\hat{ j }+\hat{ k })$

now $\overrightarrow{ b }=5 \alpha(\hat{ i }+\hat{ j }+\hat{ k })+\beta(\hat{ i })$

$\overrightarrow{ a } \cdot \overrightarrow{ b }=0$

$\Rightarrow 25 \alpha(3)+5 \beta=0$

$\Rightarrow 15 \alpha+\beta=0 \Rightarrow \beta=-15 \alpha$

$\overrightarrow{ b }=5 \alpha(-2 \hat{ i }+\hat{ j }+\hat{ k })$

$|\vec{b}|=5 \sqrt{3}$

$\Rightarrow \alpha=\pm \frac{1}{\sqrt{2}}$

$\vec{b}=\pm \frac{5}{\sqrt{2}}(-2 \hat{i}+\hat{j}+\hat{k})$

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 STD 12 - 10. vector algebra→STD 12 - 10. vector algebra — all question types→Mathematics STD 12 Science question papers→CBSE Board STD 12 Science question papers→CBSE Board question paper generator→

Similar questions

For any integer $n,$ the integral $\int_0^\pi {{e^{{{\sin }^2}x}}{{\cos }^3}(2n + 1)x\,dx = } $

If $\cos^{-1}\text{x}+\sin^{-1}\text{x}=\pi,$ then the value of $x$ is:

If $\theta=\sin^{-1}\{\sin(-600^\circ)\},$ then one of the possible values of $\theta$ is:

The area of the region bounded by parabola $y^2=x$ and the straight line $2 y=x$ is

Moving along the $ x-$ axis are two points with $x = 10 + 6t; \, x = 3 + {t^2}.$ The speed with which they are reaching from each other at the time of encounter is ........... $cm/sec$. ( $x$ is in $cm$ and $t$ is in seconds)

In a triangle $ABC,$ right angled at the vertex $A,$ if the position vectors of $A, B$ and $C$ are respectively $3\hat i\, + \hat j\, - \hat k,\,\, - \hat i\, + 3\hat j\, + p\hat k$ and $5\hat i\, + q\hat j\, - 4\hat k,\,$ then the point $(p, q)$ lies on a line

If $\frac{ dy }{ dx }+2 y \tan x =\sin x , 0< x <\frac{\pi}{2}$ and $y \left(\frac{\pi}{3}\right)=$ 0 , then the maximum value of $y(x)$ is.

If $a = i + 2j - 3k$ and $b = 3i - j + 2k,$ then the angle between the vectors $a + b$ and $a - b$ is ............... $^o$

If $\text{A}=\begin{bmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{bmatrix},$ then $A^{\top}+A=I_2$, if :

Choose the correct answer from the given four options:
The area of the region bounded by the curve $\text{y}=\sin\text{x}$ between the ordinates x = 0, $\text{x}=\frac{\pi}{2}$ and the x-axis is: