Dot Product of Two Vectors

Q.

Let $\overrightarrow{a}=\hat{i}+\hat{j}+\hat{k},\overrightarrow{b}=\hat{i}-\hat{j}+\hat{k}$ and $\overrightarrow{c}=\hat{i}-\hat{j}-\hat{k}$ be three vectors. A vector $\overrightarrow{v}$ in the plane of
$\overrightarrow{a}$ and $\overrightarrow{b}$, whose projection on $\overrightarrow{c}$ is $\frac{1}{\sqrt{3}}$, is given by

1. $\hat{i}-3\hat{j}-3\hat{k}$

2. $\hat{i}-3\hat{j}+3\hat{k}$

3. $-3\hat{i}-3\hat{j}+\hat{k}$

4. $3\hat{i}-\hat{j}+3\hat{k}$

Q. Question 1 (v)
Find
$651.2÷4$

Q. What is the difference between a theorem and an axiom?

Q.

What is the cross product of two parallel vectors?

Q.

In the figure below, ABC is a right angled triangle and M is the midpoint of AB.

(a) Prove that .

(b) Prove that MC = MA = MB.

Q.

A horizontal force acts on a body placed on a smooth horizontal table. If the force applied is $10N$, find the total work done when the displacement of the body is $5m$.

Q.

In ΔPQR, S is any point on the side QR, then

1. PQ + QR + RP < 2PS

2. PQ + QR + RP = 2PS

3. PQ + QR + RP > 2PS

4. PQ + QR + RP < 4PS

Q.

Let P, Q, R and S be the points on the plane with position vectors
$-2\hat{i}-\hat{j},4\hat{i},3\hat{i}+3\hat{j}and-3\hat{i}+2\hat{j}$, respectively. The quadrilateral PQRS must be a

1. parallelogram, which is neither a rhombus nor a rectangle

2. square

3. rhombus, but not a square

4. rectangle, but not a square

Q.

Force F acts on a body such that force F makes an angle θ with the horizontal direction and the body is also displaced through a distance s in the horizontal direction, then the work done by the forcce is

1. Fs

2. $Fs\cos \theta$

3. $Fs\sin \theta$

4. Zero

Q.

In the given figure, LM||RN and LN||RQ, where L, M, and N are points on ΔPQR. PQ = 25 cm and MN = 4 cm.

The value of PM:NQ is either

1. 1:5 or 2:3

2. 1:20 or 16:5

3. 1:4 or 3:7

4. 1:10 or 4:11

Q.

A horizontal force acts on a body placed on a smooth horizontal table. If the force applied is $10N$, find the total work done when the displacement of the body is $5m$.

In the above example if the force acts at an angle of $60°$ to the horizontal, find the work done for the same amount of displacement.

Q.

Let two non-collinear unit vectors $\hat{a}\text{and}\hat{b}$ form an acute angle. A point P moves so that at any time t the position vector
$\overrightarrow{OP}$ (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 $\overrightarrow{OP}$ and $\hat{u}$ be the unit vector along $\overrightarrow{OP}$. Then,

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

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

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

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

Q.

Use the following information to answer the next question.

The given figure shows an isosceles ΔABC with equal sides AB and AC. Side BC of the triangle is extended to point X.

What is the magnitude of ∠ACX?

1. 50°

2. 65°

3. 115°

4. 130°

Q.

$Themagnitudeoftheprojectionof2\hat{i}+3\hat{j}+4\hat{k}onthevector\hat{i}+\hat{j}+\hat{k}willbe\underset{–}{}$

1. 

2. None of the above

3. 

4. 

Q.

If $\overline{a}and\overline{b}$ are two vectors as shown in the figure, then $\overline{a}.\overline{b}$ will equal to $|\overline{a}||\overline{b}|cos\theta$

1. True

2. False

Q.

If $\overrightarrow{a}\times (\overrightarrow{b}\times \overrightarrow{c})$ is perpendicular to $(\overrightarrow{a}\times \overrightarrow{b})\times \overrightarrow{c}$ then, we may have,

1. None of these

2. $\overrightarrow{b}.\overrightarrow{c}=0$

3. $\overrightarrow{a}.\overrightarrow{b}=0$

4. $\overrightarrow{a}.\overrightarrow{c}=0$

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.

What are the possible location(s) of circumcentre in a triangle?

1. Inside the triangle

2. Outside the triangle

3. On the hypotenuse of the triangle

4. On either of the vertices of the triangle

Q.

A force of 60 N acts on a body that moves it through a distance of 4m on a horizontal surface. What is the work done, if the direction of force is at an angle of 60° to the horizontal surface?

1. 100J

2. 120J

3. 75J

4. 120N cm

Q.

If $\overrightarrow{a}=x\hat{i}+(x-1)\hat{j}+\hat{k}$ and $\overrightarrow{b}=(x+1)\hat{i}+\hat{j}+a\hat{k}$ always make an acute angle with each other for every value of $xϵR$, then

1. $aϵ(-\infty ,2)$

2. $aϵ(2,\infty )$

3. $aϵ(-\infty ,1)$

4. $aϵ(1,\infty )$

Q.

Suppose that p, q and r are three non-coplanar vectors in $R^3$. Let the components of a vector s along p, q and r be 4, 3 and 5 respectively. If the components of this vector s along (-p+q+r), (p-q+r) and (-p-q+r) are $x$, $y$ and $z$ respectively, then the value of $2x+y+z$ is ___

Q.

The angle between the vectors $\overline{u}=<3,0>and\overline{v}=<5,5>$ is ___

Q.

George is pulling Cameron on a toboggan and is exerting a force of 40N acting at an angle of ${60}^{\circ }$ to the ground. If Cameron is pulled by a distance of 100 m horizontally, then the work done by George is __ J

Q.

When the force applied and the displacement of the body are inclined at $90°$ with each other, the work done is

1. infinite

2. maximum

3. $0$

4. unity

Q.

Suppose that p, q and r are three non-coplanar vectors in $R^3$. Let the components of a vector s along p, q and r be 4, 3 and 5 respectively. If the components of this vector s along (-p+q+r), (p-q+r) and (-p-q+r) are $x$, $y$ and $z$ respectively, then the value of $2x+y+z$ is ___