Dot Product of Two Vectors

Q.

Let $R=\left\{\left(P,Q\right)\right|P,Q$ are at the same distance from the origin} , then the equivalence class of $\left(1,-1\right)$ is the set :

1. $S=\left\{\left(x,y\right)|x^2+y^2=1\right\}$

2. $S=\left\{\left(x,y\right)|x^2+y^2=4\right\}$

3. $S=\left\{\left(x,y\right)|x^2+y^2=\sqrt{2}\right\}$

4. $S=\left\{\left(x,y\right)|x^2+y^2=2\right\}$

Q.

The sum and difference of two perpendicular vectors of equal length are:

1. Perpendicular to each other and of equal length.

2. Perpendicular to each other and of different lengths.

3. Of equal length and have an obtuse angle between them.

4. Of equal length and have an acute angle between them.

Q.

What are cross product and dot product?

Q.

Let $\overrightarrow{c}$be a vector perpendicular to the vectors $a=i+j-k$and $b=i+2j+k$. If $c·(i+j+3k)=8$, then the value of $c·(axb)$ is equal to

Q.

If $\theta$ be the angle between the vectors $a=2i+2j-k$and$b=6i-3j+2k,$ then:

1. $\cos \theta =\frac{4}{21}$

2. $\cos \theta =\frac{3}{19}$

3. $\cos \theta =\frac{2}{19}$

4. $\cos \theta =\frac{5}{21}$

Q.

If $(-7-24i{)}^{\frac{1}{2}}=x-iy$, then $x^2+y^2$ is equal to

Q.

If the angle $2\theta$ is acute, then the acute angle between $x^2(\cos \theta -\sin \theta )+2xy\cos \theta +y^2(\cos \theta +\sin \theta )=0$is

1. $2\theta$

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

3. θ

4. $\frac{\theta }{2}$

Q. Let $\overrightarrow{a}=\hat{i}+\alpha \hat{j}+3\hat{k}$ and $\overrightarrow{b}=3\hat{i}-\alpha \hat{j}+\hat{k}$. If the area of the parallelogram whose adjacent sides are represented by the vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is $8\sqrt{3}$ square units, then $\overrightarrow{a}\cdot \overrightarrow{b}$ is equal to

Q. Let A vector =2i +j and B vector =3j-k and c vector =6i -2k find the value of A-2B +3c

Q. Let $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ be three non-coplanar unit vectors such that the angle between each pair of vectors is $\frac{\pi }{3}.$ If $\overrightarrow{a}\times \overrightarrow{b}+\overrightarrow{b}\times \overrightarrow{c}=p\overrightarrow{a}+q\overrightarrow{b}+r\overrightarrow{c},$ where $p,q$ and $r$ are scalars, then the value of $\frac{p^2+2q^2+r^2}{q^2}$ is

Q.

The edges of a parallelopiped are of unit length and are parallel to non-coplanar unit vectors $a,b,c$ such that $a.b=b.c=c.a=\frac{1}{2}$. Then, the volume of the parallelopiped is?

Q.

Solve for $\theta$:

$\tan \theta +cot\theta =2$

Q. Let $\overrightarrow{a}$ and $\overrightarrow{b}$ be two unit vectors such that $\overrightarrow{a}\cdot \overrightarrow{b}=0$. For some $x,y\in R$, let $\overrightarrow{c}=x\overrightarrow{a}+y\overrightarrow{b}+(\overrightarrow{a}\times \overrightarrow{b})$. If $|\overrightarrow{c}|=2$ and the vector $\overrightarrow{c}$ is inclined at the same angle $\alpha$ to both $\overrightarrow{a}$ and $\overrightarrow{b}$, then the value of $8{\cos }^2\alpha$ is .

Q.

If $\lambda (3i+2j-6k)$ is a unit vector, then the values of $\lambda$ are?

1. $\pm \frac{1}{7}$

2. $\pm 7$

3. $\pm \sqrt{43}$

4. $\pm \frac{1}{\sqrt{43}}$

5. $\pm \frac{1}{\sqrt{7}}$

Q.

The equation to the straight line passing through the point $(a{\cos }^3\theta ,a{\sin }^3\theta )$ and perpendicular to the line $xsec\theta +y\cos ec\theta =a$, is

1. $x\cos \theta -y\sin \theta =a\cos 2\theta$

2. $x\cos \theta +y\sin \theta =a\cos 2\theta$

3. $x\sin \theta +y\cos \theta =a\cos 2\theta$

4. None of these

Q.

If $\mathrm{p}\to (\mathrm{p}\wedge ~\mathrm{q})$ is false. Then the truth values of $\mathrm{p}$ and $q$ are respectively

1. F, T

2. T, F

3. F, F

4. T, T

Q. Let $\overrightarrow{a}=2\hat{i}+{\lambda }_1\hat{j}+3\hat{k},$$\overrightarrow{b}=4\hat{i}+(3-{\lambda }_2)\hat{j}+6\hat{k}$ and $\overrightarrow{c}=3\hat{i}+6\hat{j}+({\lambda }_3-1)\hat{k}$ be three vectors such that $\overrightarrow{b}=2\overrightarrow{a}$ and $\overrightarrow{a}$ is perpendicular to $\overrightarrow{c}$. Then the possible value of $({\lambda }_1,{\lambda }_2,{\lambda }_3)$ is :

1. $(1,3,1)$
2. $(\frac{1}{2},4,-2)$
3. $(-\frac{1}{2},4,0)$
4. $(1,5,1)$

Q.

For any base, show that $\log \left(1\times 2\times 3\right)=\log \left(1\right)+\log \left(2\right)+\log \left(3\right)$

Q.

$A,B,C,D$ are four points on a plane with position vectors $\text{a, b, c and d}$ respectively such that $(a-d).(b-c)=(b-d).(c-d)=0$. For $△ABC,D$ is the?

1. Incentre

2. orthocentre

3. centroid

4. None of these

Q.

If the magnitude of two vectors are 4 and 6 and the magnitude of their scalar product is 12√2 , then what is the angle between these vectors?

Q.

If $0<\theta ,\varphi <\frac{\mathrm{\pi }}{2},x=\sum _{n=0}^{\infty }{\cos }^{2n}\theta ,y=\sum _{n=0}^{\infty }{\sin }^{2n}\varphi ,$and $z=\sum _{n=0}^{\infty }{\cos }^{2n}\theta ·{\sin }^{2n}\varphi$ then:

1. $xyz=4$

2. $xy-z=\left(x+y\right)z$

3. $xy+yz+zx=z$

4. $xy+z=\left(x+y\right)z$

Q.

Is cross product distributive over dot product?

Q.

Let two non-collinear unit vectors $\hat{a}$ and $\hat{b}$ form an acute angle. A point $P$ moves so that at any time $t$ the position vector $OP$ (where $O$ is the origin) is given by $\hat{a}\cos t+\hat{b}\sin t$. When $P$ is farthest from origin $O$, let $M$ be the length of $OP$ and $\hat{u}$ be the unit vector along vector $OP$. Then:

1. $u=\frac{\hat{a}+\hat{b}}{\left|\hat{a}+\hat{b}\right|}$ and $M={\left(1+\hat{a}·\hat{b}\right)}^{\frac{1}{2}}$

2. $u=\frac{\hat{a}-\hat{b}}{\left|\hat{a}-\hat{b}\right|}$ and $M={\left(1+\hat{a}·\hat{b}\right)}^{\frac{1}{2}}$

3. $u=\frac{\hat{a}+\hat{b}}{\left|\hat{a}+\hat{b}\right|}$ and $M={\left(1+2\hat{a}·\hat{b}\right)}^{\frac{1}{2}}$

4. $u=\frac{\hat{a}-\hat{b}}{\left|\hat{a}-\hat{b}\right|}$ and $M={\left(1+2\hat{a}·\hat{b}\right)}^{\frac{1}{2}}$

Q.

How can we calculate inverse tan for determining angle between vectors using Natural tangents table?

Q. The projection of the line segment joining the points $(1,-1,3)$ and $(2,-4,11)$ on the line joining the points $(-1,2,3)$ and $(3,-2,10)$ is

Q. Let $\overrightarrow{a}=\hat{i}+\hat{j}+\hat{k}$ and $\overrightarrow{b}=\hat{j}-\hat{k}.$ If $\overrightarrow{c}$ is a vector such that $\overrightarrow{a}\times \overrightarrow{c}=\overrightarrow{b}$ and $\overrightarrow{a}\cdot \overrightarrow{c}=3,$ then $\overrightarrow{a}\cdot (\overrightarrow{b}\times \overrightarrow{c})$ is equal to

1. $6$
2. $-2$
3. $-6$
4. $2$

Q.

Let O be the origin and let PQR be an arbitrary triangle. The point S is such that OP.OQ + OR.OS = OR.OP + OQ.OS = OQ.OR + OP.OS Then the triangle PQR has S as its

1. centroid

2. orthocenter

3. incentre

4. circumcentre

Q. If $\overrightarrow{a}$ and $\overrightarrow{b}$ are two unit vectors such that $\overrightarrow{a}+\overrightarrow{b}$ is also a unit vector, then find the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. [CBSE 2014]

Q. The projections of a directed line segment on the coordinate axes are $12,4\text{and}3.$ The direction cosines of the line are

1. $(\frac{-12}{13},\frac{-4}{13},\frac{3}{13})$
2. $(\frac{12}{13},\frac{-4}{13},\frac{3}{13})$
3. $(\frac{12}{13},\frac{4}{13},\frac{3}{13})$
4. $(\frac{12}{13},\frac{4}{13},\frac{-3}{13})$

Q. If Lines $OA,OB$ are drawn from $O$(origin) with direction cosines proportional to $(1,-2,-1),(3,-2,3)$. Then the direction cosines of the normal to the plane $AOB$ is

1. $(-\frac{1}{\sqrt{29}},-\frac{2}{2\sqrt{29}},\frac{4}{\sqrt{29}})$
2. $(\frac{2}{\sqrt{29}},-\frac{3}{2\sqrt{29}},\frac{4}{\sqrt{29}})$
3. $(-\frac{3}{\sqrt{29}},-\frac{4}{2\sqrt{29}},\frac{2}{\sqrt{29}})$
4. $(-\frac{4}{\sqrt{29}},-\frac{3}{\sqrt{29}},\frac{2}{\sqrt{29}})$