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#### Applications of Vector Algebra Book Back Questions

12th Standard EM

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Maths

Time : 00:45:00 Hrs
Total Marks : 30
5 x 1 = 5
1. If $\vec{a}$ and $\vec{b}$ are parallel vectors, then $[\vec { a } ,\vec { c } ,\vec { b } ]$ is equal to

(a)

2

(b)

-1

(c)

1

(d)

0

2. If $\vec { a }$ and $\vec { b }$ are unit vectors such that $[\vec { a } ,\vec { b },\vec { a } \times \vec { b } ]=\frac { \pi }{ 4 }$, then the angle between $\vec { a }$ and $\vec { b }$ is

(a)

$\frac { \pi }{ 6 }$

(b)

$\frac { \pi }{ 4 }$

(c)

$\frac { \pi }{ 3 }$

(d)

$\frac { \pi }{ 2 }$

3. If $\vec { a } ,\vec { b } ,\vec { c }$ are three non-coplanar vectors such that $\vec { a } \times (\vec { b } \times \vec { c } )=\frac { \vec { b } +\vec { c } }{ \sqrt { 2 } }$, then the angle between

(a)

$\frac { \pi }{ 2 }$

(b)

$\frac { 3\pi }{ 6 }$

(c)

$\frac { \pi }{ 4 }$

(d)

${ \pi }$

4. If $\vec { a } \times (\vec { b } \times \vec { c } )=(\vec { a } \times \vec { b } )\times \vec { c }$ where $\vec { a } ,\vec { b } ,\vec { c }$ are any three vectors such that $\vec { a } ,\vec { b }$ $\neq$ 0 and  $\vec { a } .\vec { b }$ $\neq$ 0 then $\vec { a }$ and $\vec { c }$ are

(a)

perpendicular

(b)

parallel

(c)

inclined at an angle $\frac{\pi}{3}$

(d)

inclined at an angle  $\frac{\pi}{6}$

5. The angle between the line $\vec { r } =(\hat { i } +2\hat { j } -3\hat { k } )+t(2\hat { i } +\hat { j } -2\hat { k } )$ and the plane $\vec { r } .(\hat { i } +\hat { j } )+4=0$ is

(a)

(b)

30°

(c)

45°

(d)

90°

6. 3 x 2 = 6
7. Find the volume of the parallelepiped whose coterminous edges are represented by the vectors $-6\hat { i } +14\hat { j } +10\hat { k } ,14\hat { i } -10\hat { j } -6\hat { k }$ and $2\hat { i } +4\hat { j } -2\hat { k }$

8. Show that the lines $\frac { x-1 }{ 4 } =\frac { 2-y }{ 6 } =\frac { z-4 }{ 12 }$ and $\frac { x-1 }{ 4 } =\frac { 2-y }{ 6 } =\frac { z-4 }{ 12 }$ are parallel.

9. Find a parametric form of vector equation of a plane which is at a distance of 7 units from the origin having 3, −4,5 as direction ratios of a normal to it.

10. 3 x 3 = 9
11. If D is the midpoint of the side BC of a triangle ABC, then show by vector method that ${ \left| \vec { AB } \right| }^{ 2 }+{ \left| \vec { AC } \right| }^{ 2 }=2({ \left| \vec { AD} \right| }^{ 2 }+{ \left| \vec { BD } \right| }^{ 2 })$

12. Prove by vector method that the median to the base of an isosceles triangle is perpendicular to the base.

13. Find the magnitude and direction cosines of the torque of a force represented by $\hat { 3i } +\hat { 4j } -\hat { 5k }$ about the point with position vector $\hat { 2i } -\hat { 3j } +\hat { 4k }$ acting through a point whose position vector is $\hat { 4i } +\hat { 2j } -\hat { 3k }$.

14. 2 x 5 = 10
15. If $\vec { a } =-2\hat { i } +3\hat { j } -2\hat { k } ,\vec { b } =3\hat { i } -\hat { j } +3\hat { k } ,\vec { c } =2\hat { i } -5\hat { j } +\hat { k }$ find $(\vec { a } \times \vec { b } )\times \vec { c }$ and $(\vec { a } \times \vec { b } )\times \vec { c }$. State whether they are equal.

16. Find the parametric form of vector equation, and Cartesian equations of the plane containing the line $\vec { r } =(\hat { i } -\hat { j } +3\hat { k } )+t(2\hat { i } -\hat { j } +4\hat { k } )$ and perpendicular to plane $\vec { r } .(\hat { i } +2\hat { j } +\hat { k } )=8$