Position Vector

Q. A bird moves from point $A$ $(1,-2,3)$ to $B$ $(4,2,3)$. If the speed of the bird is $10\text{m/s}$, then the velocity vector of the bird is:

1. $5(\hat{i}-2\hat{j}+3\hat{k})\text{m/s}$
2. $(6\hat{i}+8\hat{j})\text{m/s}$
3. $5(4\hat{i}+2\hat{j}+3\hat{k})\text{m/s}$
4. $(0.6\hat{i}+0.8\hat{j})\text{m/s}$

Q. Ship $A$ is sailing towards north-east with velocity $\overrightarrow{v}=30\hat{i}+50\hat{j}km/h$, where $\hat{i}$ points east and $\hat{j}$ points north. Ship $B$ at a distance of $80km$ east and $150km$ north of ship $A$ is sailing towards the west at $10km/h$. $A$ will be at minimum distance from $B$ in

1. $4.2h$
2. $2.2h$
3. $2.6h$
4. $3.2h$

Q. The coordinates of the positions of particles of mass 7, 4 and 10 gm (1, 5, -3), (2, 5, 7) and (3, 3, -1) cm respectively. The position of the centre of mass of the system would be

1. 
2. 
3. 
4. 

Q.

A rabbit runs across a parking lot on which a set of coordinate axes has, strangely enough, been drawn.
The coordinates(meters) of the rabbit's position as function of time t(seconds) are given by

$x\left(t\right)=-\frac{t^2}{2}+5t+20Andy\left(t\right)=t^2-10t+30$

(a)At t=10s, what is the rabbot's position vector $\overrightarrow{r}$ in unit-vector notation and in magnitude-angle notation?

1. 

2. 

3. 

4. 

Q. At time $t=0\text{s}$, a particle starts from rest at position $(4\text{m},8\text{m})$. It starts moving towards the positive $x$-axis with a constant acceleration of $2{\text{m/s}}^2$. After $2\text{s}$, it undergoes an additional acceleration of $4{\text{m/s}}^2$ in the positive $y$-direction. Find the coordinates of the particle (in $\text{m}$) after the next $5\text{s}$.

1. $(58,6)$
2. $(-53,58)$
3. $(53,58)$
4. $(58,-6)$

Q. The position vector of point B (8, 4) with respect to point A (3, 7) is

1. 8i + 4j
2. 3i + 7j
3. 11i + 11j
4. 5i – 3j

Q. A bird moves from point $A$ $(1,-2,3)$ to $B$ $(4,2,3)$. If the speed of the bird is $10\text{m/s}$, then the velocity vector of the bird is:

1. $5(\hat{i}-2\hat{j}+3\hat{k})\text{m/s}$
2. $5(4\hat{i}+2\hat{j}+3\hat{k})\text{m/s}$
3. $(0.6\hat{i}+0.8\hat{j})\text{m/s}$
4. $(6\hat{i}+8\hat{j})\text{m/s}$

Q. Point $(4,0)$ lies on ________.

1. $\overrightarrow{YO}$
2. $\overrightarrow{XO}$
3. $\overrightarrow{OX}$
4. $\overrightarrow{OY}$

Q. Two cars $A$ and $B$ move on a straight road with constant velocity. When they move in same direction, they come close by $8\text{m}$ in $10\text{s}$ and when they move in opposite direction towards each other, they come close by $20\text{m in 1 s}$. The speed of the cars A and B (in $m/s$) respectively are

1. $9.6\text{and}10.4$
2. $10.4\text{and}9.6$
3. $20.8\text{and}10.4$
4. $10.4\text{and}20.8$

Q. Two cars $A$ and $B$ move on a straight road with constant velocity. When they move in same direction, they come close by $8\text{m}$ in $10\text{s}$ and when they move in opposite direction towards each other, they come close by $20\text{m in 1 s}$. The speed of the cars A and B (in $m/s$) respectively are

1. $9.6\text{and}10.4$
2. $10.4\text{and}9.6$
3. $20.8\text{and}10.4$
4. $10.4\text{and}20.8$

Q. A particle follows uniform motion along line from P $(2,4,6)$ to Q$(4,8,2)$ with a speed of $12$ m/s. Find velocity vector of the particle.

1. $2\hat{i}+4\hat{j}-4\hat{k}$
2. none
3. $12(2\hat{i}+4\hat{j}-4\hat{k})$
4. $4\hat{i}+8\hat{j}-8\hat{k}$

Q. Two trains $\text{A}$ and $\text{B}$ have lengths $800\text{m}$ and $1000\text{m}$ respectively and are moving with the velocity of $25\text{m/s}$ and $15\text{m/s}$ respectively. At time $t=0$, engine of train $\text{A}$ is just behind tail of train $\text{B}$. Find the time when train $\text{A}$ completely overtake train $\text{B}?$

1. $10\text{sec}$
2. $100\text{sec}$
3. $18\text{sec}$
4. $180\text{sec}$

Q. Ship $A$ is sailing towards north-east with velocity $\overrightarrow{v}=30\hat{i}+50\hat{j}km/h$, where $\hat{i}$ points east and $\hat{j}$ points north. Ship $B$ at a distance of $80km$ east and $150km$ north of ship $A$ is sailing towards the west at $10km/h$. $A$ will be at minimum distance from $B$ in

1. $4.2h$
2. $2.6h$
3. $3.2h$
4. $2.2h$

Q.

Find the resultant of the given vectors.

1. 

2. 

3. 

4. Cannot add as is not on origin

Q. A wheel of radius $r$ rolls without slipping on the ground, with speed $v$. When it is at a point $P$, a piece of mud flies off tangentially from its highest point, lands on the ground at point $Q$. The distance $PQ$ is:

1. $2v\sqrt{\left(r/g\right)}$
2. $2\sqrt{2}v\sqrt{\left(r/g\right)}$
3. $4v\sqrt{\left(r/g\right)}$
4. $v\sqrt{\left(r/g\right)}$

Q. A particle moves along the x-axis with acceleration a = 6(t- 1), where t is in seconds. If the particle is initially at the origin and moves along the positive x-axis with $v_0$ = 2 m/s, analyze the motion of the particle.

Q. If $\overline{p}=\hat{i}-2\hat{j}+\hat{k}$ and $\overline{q}=\hat{i}+4\hat{j}-2\hat{k}$ are position vectors of points $P$ and $Q$, find the position vector of the point $R$ which divides segment $PQ$ internally in the ratio $2:1$

Q. A particle starts from rest at the origin and moves along X-axis with acceleration $a=12-2t$. The time after which the particle arrives at the origin is (in seconds)

1. $12$
2. $17$
   
3. $18$
4. $14$
   

Q. The position vectors of the points $A,B,C$ and $D$ are $3\hat{i}-2\hat{j}-\hat{k},2\hat{i}+3\hat{j}-4\hat{k},-\hat{i}+\hat{j}+2\hat{k}$ and $4\hat{i}+5\hat{j}+\lambda \hat{k}$ respectively. If the points $A,B,C$ and $D$ lie on a plane, find the value of $\lambda$.

Q. Two identical particles are located at $\overrightarrow{x}$ and $\overrightarrow{y}$ with reference to the origin of three dimensional co-ordinate system. The position vector of centre of mass of the system is given by

1. $\overrightarrow{x}-\overrightarrow{y}$
2. $\frac{\overrightarrow{x}+\overrightarrow{y}}{2}$
3. $\frac{\overrightarrow{x}-\overrightarrow{y}}{2}$
4. $(\overrightarrow{x}-\overrightarrow{y})$

Q. If $C$ is the midpoint of $AB$ and $P$ is any point outside $AB$, then $\overrightarrow{PA}+\overrightarrow{PB}=$

Q. If the position vectors of $A,B,C,D$ are $3\hat{i}+2\hat{j}+\hat{k},4\hat{i}+5\hat{j}+5\hat{k},4\hat{i}+2\hat{j}-2\hat{k},6\hat{i}+5\hat{j}-\hat{k}$ respectively then the position vector of the point of intersection of $\overline{AB}$ and $\overline{CD}$ is

1. $2\hat{i}+\hat{j}-3\hat{k}$
2. $2\hat{i}-\hat{j}+3\hat{k}$
3. $2\hat{i}+\hat{j}+3\hat{k}$
4. $2\hat{i}-\hat{j}-3\hat{k}$

Q. At time $t=0\text{s}$, a particle starts from rest at position $(4\text{m},8\text{m})$. It starts moving towards the positive $x$-axis with a constant acceleration of $2{\text{m/s}}^2$. After $2\text{s}$, it undergoes an additional acceleration of $4{\text{m/s}}^2$ in the positive $y$-direction. Find the coordinates of the particle (in $\text{m}$) after the next $5\text{s}$.

1. $(58,6)$
2. $(53,58)$
3. $(-53,58)$
4. $(58,-6)$

Q.

A watermelon seed has the following coordinates: x=-5.0 m, y=9.0 m and z=0 m. Find its position vector

(a) In unit-vector notation and as
(b) A magnitude and
(c) An angle relative to the positive direction of the x-axis

1. 

2. 

3. 

4. 

Q. The $P.V{.}^{\prime }s$ of the vertices of a $△ABC$ are $\overline{i}+\overline{j}+\overline{k},4\overline{i}+\overline{j}+\overline{k},4\overline{i}+5\overline{j}+\overline{k}$. The $P.V.$ of the circumcentre of $△ABC$ is

1. $\frac{5}{2}\overline{i}+3\overline{j}+\overline{k}$
2. $5\overline{i}+\frac{3}{2}\overline{j}+\overline{k}$
3. $\overline{i}+\overline{j}+\overline{k}$
4. $5\overline{i}+3\overline{j}+\frac{1}{2}\overline{k}$

Q. The position vectors of $P$ and $Q$ are respectively $a$ and $b$. If $R$ is a point on $PQ$, $PQ$ such that $PR=5PQ$, then the position vector of $R$ is

1. $5b+4a$
2. $4b-5a$
3. $5b-4a$
4. $4b+5a$

Q. If $\hat{i},\hat{j},\hat{k}$ are positive vectors of $A,B,C$ and $\overrightarrow{AB}=\overrightarrow{CX}$, then positive vector of $X$ is

1. $-\hat{i}+\hat{j}+\hat{K}$
2. $\hat{i}-\hat{j}+\hat{K}$
3. $\hat{i}+\hat{j}-\hat{K}$
4. $\hat{i}+\hat{j}+\hat{K}$

Q. Find the position vectors of a point $R$ which divides the line joining two points $P$ and $Q$ whose position vectors are $2\overrightarrow{a}+3\overrightarrow{b}$ and $\overrightarrow{a}-3\overrightarrow{b}$ respectively, externally in the ratio $1:2$. Also, show that $P$ is the midpoint of the line segment $R$.

Q. If $\overrightarrow{b}$ and $\overrightarrow{c}$ are the position vectors of the points B and C respectively, then the position vector of the point D such that $\overrightarrow{BD}=4\overrightarrow{BC}$ is

1. $4(\overrightarrow{c}-\overrightarrow{b})$
2. $4\overrightarrow{c}-3\overrightarrow{b}$
3. $-4(\overrightarrow{c}-\overrightarrow{b})$
4. $4\overrightarrow{c}+3\overrightarrow{b}$

Q. L and M are two points with position vectors $2\overrightarrow{a}-\overrightarrow{b}$ and $\overrightarrow{a}+2\overrightarrow{b}$ respectively. Write the position vector of a point N which divides the line segment LM in the ratio 2 : 1 externally.