Vector Algebra Test

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Vector Algebra Test - 72

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Question 1
1 / -0

Given unit vectors $$\hat {m}, \hat {n}$$ and $$\hat {p}$$ such that $$\left( \widehat { \hat { m } \hat { n } } \right) =\hat { p } \widehat { } \left( \hat { m } \times \hat { n } \right) =\alpha$$ then the value of $$[\hat { n } \hat { p } \hat { m } ]$$ in terms of $$\alpha$$ is :

A
$$\sin \alpha$$

B
$$\cos \alpha$$

C
$$\sin \alpha \cos \alpha $$

D
$$\sin^{2} \alpha$$

Solution
Question 2
1 / -0

The position vector of two points A and B are 6a+2b and a-3b. If a point C divides AB in the ratio 3 : 2, then the position vector of C is

A
3a-b

B
3a+b

C
a+b

D
a-b

Solution
Question 3
1 / -0

$$\left| {\overline x } \right| = \left| {\overline y } \right| = 1,\,\overline x \bot \overline y ,\,\left| {\overline x + \overline y } \right| = $$

A
$$\sqrt 3 $$

B
$$\sqrt 2 $$

C
$$1$$

D
$$0$$

Solution
Question 4
1 / -0

The position vector of a point $$C$$ with respect to $$B$$ is $$\hat { i } + \hat { j }$$ and that of B with respect to A is $$\hat { i } - \hat { j }$$. The position vector of $$C$$ with respect to $$A$$ is

A
$$\hat { 2i }$$

B
$$-\hat { 2i }$$

C
$$\hat { 2j }$$

D
$$-\hat { 2j }$$
Question 5
1 / -0

D,E and F are the mid-points of the sides BC,CA and AB respectively of $$\Delta ABC$$ and G is the centroid of the triangle, then $$\vec{GD}+\vec{GE}+\vec{GF}=$$

A
$$\vec{0}$$

B
$$2\vec{AB}$$

C
$$2\vec{GA}$$

D
$$2\vec{GC}$$

Solution
Question 6
1 / -0

If $$\vec { u } =\vec { a } -\vec { b } ~;\vec{ v } =\vec { a } +\vec { b } ~~\&~ |\vec { a }| =|\vec { b }| =2,$$ then $$\left| \vec { u } \times \vec { \upsilon } \right| $$ is equal to:

A
$$\sqrt { 2\left( 4-\left( \vec { a } .\vec { b } \right) ^{ 2 } \right) } $$

B
$${ 2 }\sqrt { \left( 16+\left( \vec { a } .\vec { b } \right) ^{ 2 } \right) } $$

C
$${ 2 }\sqrt { \left( 4-\left( \vec { a } .\vec { b } \right) ^{ 2 } \right) } $$

D
$${ 2 }\sqrt { \left( 16-\left( \vec { a } .\vec { b } \right) ^{ 2 } \right) } $$

Solution
Question 7
1 / -0

Vector equation of the plane $$\vec{r}=\hat{i}-\hat{j}+\lambda(\hat{i}+\hat{j}+\hat{k})+\mu(\hat{i}+2\hat{j}+3\hat{k})$$ in the scalar dot product from is

A
$$\vec{r}.(5\hat{i}-2\hat{j}+3\hat{k})=7$$

B
$$\vec{r}.(5\hat{i}-2\hat{j}-3\hat{k})=7$$

C
$$\vec{r}.(5\hat{i}+2\hat{j}-3\hat{k})=7$$

D
$$\vec{r}.(5\hat{i}+2\hat{j}+3\hat{k})=7$$

Solution
Question 8
1 / -0

$$\vec{r}.\hat{i}=2\vec{r}.\hat{j}=4\vec{r}.\hat{k}$$ and $$\left|\vec{r}\right|=\sqrt{84}$$, then $$\left|\vec{r}.\left(2\hat{i}-3\hat{j}+\hat{k}\right)\right|$$ is equal to

A
$$0$$

B
$$2$$

C
$$4$$

D
$$6$$

Solution
Question 9
1 / -0

If $$a,b$$ and $$c$$ are position vector of $$A,B$$ and $$C$$ respectively of $$\triangle ABC$$ and if $$|a-b|=4,|b-c|=2, |c-a|=3$$, then the distance between the centroid and incentre of $$\triangle ABC$$ is

A
$$1$$

B
$$\dfrac{1}{2}$$

C
$$\dfrac{1}{3}$$

D
$$\dfrac{2}{3}$$

Solution
Question 10
1 / -0

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 $$\bar{AB}$$ and $$\bar{CD}$$ is

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

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

C
$$2\hat{i} + \hat{j} + 3\hat{k}$$

D
$$2\hat{i} - \hat{j} - 3\hat{k}$$

Solution

$$ \begin{aligned} \overrightarrow{A B} &=\vec{b}-\vec{a} \\ &=(4 i+5 \hat{\jmath}+5 \hat{k})-(3 \hat{i}+2 \hat{\jmath}+\hat{k}) \\ &=\hat{\imath}+3 \hat{\jmath}+4 \hat{k} , and \end{aligned} $$
$$\begin{aligned} \overrightarrow{C D} &=\vec{d}-\vec{c} \\ &=(6 \hat{\imath}+5 \hat{\jmath}-\hat{k})-(4 \hat{\imath}+2 \hat{\jmath}-2 \hat{k}) \\ &=\overrightarrow{2} \hat{\imath}+3 \hat{\jmath}+\hat{k} \\ \text { Let } \overrightarrow{A B} & \text { and } \overrightarrow{C D} \text { intersect at } \vec{P} . \\ \text { Line } A B=(3 i+2 j+\hat{k})+\lambda(i+3 \hat{j}+4 \hat{k}) & \text { and } \end{aligned}$$
Line $$C D=(4 i+2 \hat{\imath}-2 \hat{k})+\mu(2 \hat{i}+3 \hat{j}+\hat{k})$$
W.r.t. line $$A_{0}$$,
$$P=(3+\lambda, 2+3 \lambda, 1+4 \lambda)$$
and $$w \cdot r \cdot t \cdot$$ line $$C D$$,
$$P=(4+2 \mu, 2+3 \mu,-2+\mu)$$
Comparing both points,
$$\begin{aligned} \Rightarrow & 3+\lambda^{\prime}=4+2 \mu, \\ & 2+3 \lambda=2+3 \mu \\ & 1+4 \lambda=-2+\mu \end{aligned}$$
Solving these equs.,
$$\therefore \vec{p}=2 \hat{\imath}-\hat{\jmath}-3 \hat{k}$$
$$\therefore$$ option $$D$$ is correct.