Applications of Dot Product

Q. Let $\overrightarrow{a}=\hat{i}+\hat{j}+\sqrt{2}\hat{k},\overrightarrow{b}=b_1\hat{i}+b_2\hat{j}+\sqrt{2}\hat{k}$ and $\overrightarrow{c}=5\hat{i}+\hat{j}+\sqrt{2}\hat{k}$ be three vectors such that the projection vector of $\overrightarrow{b}$ on $\overrightarrow{a}$ is $\overrightarrow{a}$. If $\overrightarrow{a}+\overrightarrow{b}$ is perpendicular to $\overrightarrow{c}$, then $|\overrightarrow{b}|$ is equal to :

1. $\sqrt{32}$
2. $4$
3. $6$
4. $\sqrt{22}$

Q.

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

1. $60°$

2. $90°$

3. $45°$

4. $55°$

Q.

Let $\stackrel{\rightharpoonup }{a}=\stackrel{\rightharpoonup }{i}+2\stackrel{\rightharpoonup }{j}-3\stackrel{\rightharpoonup }{k}$ and $\stackrel{\rightharpoonup }{b}=2\stackrel{\rightharpoonup }{i}-3\stackrel{\rightharpoonup }{j}+5\stackrel{\rightharpoonup }{k}$. If $\stackrel{\rightharpoonup }{r}\times \stackrel{\rightharpoonup }{a}=\stackrel{\rightharpoonup }{b}\times \overrightarrow{r}$, $\stackrel{\rightharpoonup }{r}.\left(a\stackrel{\rightharpoonup }{i}+2\stackrel{\rightharpoonup }{j}+\stackrel{\rightharpoonup }{k}\right)=3$ and $\stackrel{\rightharpoonup }{r}.\left(2\stackrel{\rightharpoonup }{i}+5\stackrel{\rightharpoonup }{j}-a\stackrel{\rightharpoonup }{k}\right)=-1$, $ɑ\in R$, then the value of $a+{\left|\stackrel{\rightharpoonup }{r}\right|}^2=$

1. $11$

2. $15$

3. $9$

4. $13$

Q.

The value of $\cos \left(12°\right)+\cos \left(84°\right)+\cos \left(156°\right)+\cos \left(132°\right)$

1. $\frac{1}{2}$

2. $1$

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

4. $\frac{1}{8}$

Q.

If vectors$a_1=x\hat{i}-\hat{j}+\hat{k}$ and $a_2=\hat{i}+y\hat{j}+z\hat{k}$ are collinear, then a possible unit vector parallel to the vector$x\hat{i}+y\hat{j}+z\hat{k}$ is

1. $\left(\frac{1}{\sqrt{2}}\right)\left(-\hat{j}+\hat{k}\right)$

2. $\left(\frac{1}{\sqrt{2}}\right)\left(\hat{i}–\hat{j}\right)$

3. $\left(\frac{1}{\sqrt{3}}\right)\left(\hat{i}-\hat{j}+\hat{k}\right)$

4. $\left(\frac{1}{\sqrt{3}}\right)\left(\hat{i}+\hat{j}–\hat{k}\right)$

Q.

The length of perpendicular from $(3,1)$ on line $4x+3y+20=0$, is

1. $6$

2. $7$

3. $5$

4. $8$

Q.

In a “magic square”, the sum of the numbers in each row, in each column and along the diagonals is the same. Is this a magic square?

$\frac{4}{11}$	$\frac{9}{11}$	$\frac{2}{11}$
$\frac{3}{11}$	$\frac{5}{11}$	$\frac{7}{11}$
$\frac{8}{11}$	$\frac{1}{11}$	$\frac{6}{11}$

Q.

The distance between the lines $3x+4y=9$ and $6x+8y=15$ is:

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

2. $\frac{3}{10}$

3. $6$

4. None of these

Q.

If $\sec \left(\theta \right)+\tan \left(\theta \right)=p$, then $\tan \left(\theta \right)$ is equal to

1. $\frac{2p}{p^2-1}$

2. $\frac{p^2-1}{2p}$

3. $\frac{p^2+1}{2p}$

4. $\frac{2p}{p^2+1}$

Q.

Find the principal value of ${\cot }^{-1}\left(\sqrt{3}\right)$

Q.

If $\frac{(x+y)}{2},y,\frac{(y+z)}{2}$ are in HP, then $x,y,z$ are in

1. AP

2. GP

3. HP

4. None of these

Q.

If $a,b$and $c$ are three vectors such that $a\times b=c$ and $b\times c=a$, then

1. $a\not\equiv b\not\equiv c$

2. $a=b=c$

3. $a\not\equiv b\not\equiv c\not\equiv 1$

4. $a,b$and $c$ are orthogonal in pairs.

Q. If a+b+c=0 and |a|=3, |b|=4 and $|c|=\sqrt{37}$, the angle between a and b is

1. $\frac{\pi }{3}$
2. $\frac{\pi }{4}$
3. $\frac{\pi }{2}$
4. $\frac{\pi }{6}$

Q.

Forces of magnitudes $5$ and$3$ units acting in the directions $6i+2j+3k$ and $3i-2j+6k$respectively act on a particle which is displaced from the point $(2,2,–1)$ to $(4,3,1)$. Find the work done by the forces.

1. $127/7units$

2. $148/7units$

3. $133/7units$

4. $139/7units$

Q.

What does a dot product equal to $1$ mean?

Q. If the vector $\overrightarrow{OP}=\hat{i}+2\hat{j}+2\hat{k}$ rotates through a right angle about origin, passing through the positive $x-$axis on the way becomes $\overrightarrow{OQ}=x\hat{i}+y\hat{j}+z\hat{k}$, then the value of $x-y+z$ is

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

Q.

If the vectors $a=i+aj+a^{2}k,b=i+bj+b^{2}k,$ and $c=i+cj+c^{2}k$ are three non-coplanar vectors and $\left|\begin{array}{ccc}a& a^2& 1+a^3\\ b& b^2& 1+b^3\\ c& c^2& 1+c^3\end{array}\right|=0$, then the value of $abc$ is

1. $0$

2. $1$

3. $2$

4. $-1$

Q.

If $A=\{x\in R:|x|<2\}$and $B=\{x\in R:|x-2|\ge 3\}$ then

1. $A–B=[-1,2]$

2. $B–A=R-(-2,5)$

3. $A\cup B=R-(2,5)$

4. $A\cap B=(-2,-1)$

Q.

A ladder is resting with the wall at an angle of $30$ degrees. A man is ascending the ladder at the rate of $3$ ft/sec.

His rate of approaching the wall is

1. $3$ ft/sec

2. $\frac{3}{2}$ ft/sec

3. $\frac{2}{4}$ ft/sec

4. $\frac{3}{\sqrt{2}}$ ft/sec

Q.

The value of $\cos \left(270°+\theta \right)\cos \left(90°-\theta \right)-\sin \left(270°-\theta \right)\cos \left(\theta \right)$ is

1. $0$

2. $-1$

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

4. $1$

Q.

The straight lines $x+2y-9=0,3x+5y-5=0$ and $ax+by-1=0$ are concurrent, if the straight line $35x-22y+1=0$ passes through the point

1. $(a,b)$

2. $(b,a)$

3. $(-a,-b)$

4. None of these

Q. The line of shortest distance between the lines $\frac{x-2}{0}=\frac{y-1}{1}=\frac{z}{1}\text{and}\frac{x-3}{2}=\frac{y-5}{2}=\frac{z-1}{1}$ makes an angle of ${\cos }^{-1}\left(\sqrt{\frac{2}{27}}\right)$ with the plane $P:ax-y-$ $z=0,(a>0)$. If the image of the point $(1,1,-5)$ in the plane $P\text{is}(\alpha ,\beta ,\gamma )$, then $\alpha +\beta -\gamma$ is equal to

Q.

Find the sine of the angle between the vectors $\overrightarrow{a}=3\hat{i}+\hat{j}+2\hat{k}and\overrightarrow{b}=2\hat{i}-2\hat{j}+4\hat{k}$.

Q.

If the acute angle between the two regression lines is $\theta$, then

1. $\sin \theta \ge 1–r^2$

2. $\tan \theta \ge 1–r^2$

3. $Sin\theta \le 1–r^2$

4. $\tan \theta \le 1–r^2$

Q.

If $\left|a\right|=1$, $\left|b\right|=4,a.b=2$ and $c=2a\times b–3b$, then the angle between $b$ and $c$ is?

1. $\frac{\mathrm{\pi }}{6}$

2. $\frac{5\mathrm{\pi }}{6}$

3. $\frac{\mathrm{\pi }}{3}$

4. $\frac{2\mathrm{\pi }}{3}$

Q.

Two sides of a triangle are given by the roots of the equations $x^2-5x+6=0$ and the angle between the sides is $\frac{\pi }{3}$ Then, the perimeter of the triangle is

1. $5+\sqrt{2}$

2. $5+\sqrt{3}$

3. $5+\sqrt{5}$

4. $5+\sqrt{7}$

Q.

Find the value of$\log 1$ to the base $3$ ?

Q.

If the vectors $a,b,$ and $c$ from the sides $BC,CA,$ and $AB,$ respectively of a $△ABC$s, then

1. $a·b+b·c+c·a=0$

2. $a\times b=b\times c=c\times a$

3. $a·b=b·c=c·a$

4. $a\times b+b\times c+c\times a=0$

Q. The magnitude of the projection of the vector $2\hat{i}+3\hat{j}+\hat{k}$ on the vector perpendicular to the plane containing the vectors $\hat{i}+\hat{j}+\hat{k}$ and $\hat{i}+2\hat{j}+3\hat{k},$ is:

1. $3\sqrt{6}$
2. $\frac{\sqrt{3}}{2}$
3. $\sqrt{6}$
4. $\sqrt{\frac{3}{2}}$

Q.

If $a\&b$ are any two vectors, then $(2a+3b)\times (5a+7b)+a\times b$ is equal to?

1. $0$

2. $1$

3. $a\times b$

4. $b\times a$