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#### Applications of Vector Algebra - Two Marks Study Materials

12th Standard EM

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Maths

Time : 01:00:00 Hrs
Total Marks : 30
15 x 2 = 30
1. Show that the vectors $\hat { i } +\hat { 2j } -\hat { 3k }$,$\hat { i } +\hat { 2j } -\hat { 3k }$ and $\hat { 3i } +\hat { j } -\hat { k }$

2. Determine whether the three vectors $2\hat { i } +3\hat { j } +\hat { k }$$\hat { i } -2\hat { j } +2\hat { k }$ and $\hat { 3i } +\hat { j } +2\hat { k }$ are coplanar.

3. Find the non-parametric form of vector equation and Cartesian equations of the straight line passing through the point with position vector  $4\hat { i } +3\hat { j } -7\hat { k }$ and parallel to the vector $2\hat { i } -6\hat { j } +7\hat { k }$.

4. Find the angle between the line $\vec { r } =(2\hat { i } -\hat { j } +\hat { k } )+t(\hat { i } +2\hat { j } -2\hat { k } )$ and the plane $\vec { r } =(6\hat { i } +3\hat { j } +2\hat { k } )=8$

5. Find the angle between the following lines.
2x = 3y =  −z and 6x = − y = −4z.

6. If $\overset { \rightarrow }{ a } =\overset { \wedge }{ i } +2\overset { \wedge }{ j } +3\overset { \wedge }{ k }$$\overset { \rightarrow }{ b } =-\overset { \wedge }{ i } +2\overset { \wedge }{ j } +\overset { \wedge }{ k }$ and $\overset { \rightarrow }{ c } =3\overset { \wedge }{ i } +\overset { \wedge }{ j }$ find $\frac { \lambda }{ c }$ such that $\overset { \rightarrow }{ a } +\lambda \overset { \rightarrow }{ b }$ is perpendicular to $\overset { \rightarrow }{ c }$

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-a

7. A force of magnitude 6 units acting parallel to $\overset { \wedge }{ 2i } -\overset { \wedge }{ 2j } +\overset { \wedge }{ k }$ displaces the point of application from (1, 2, 3) to (5, 3, 7). Find the work done.

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b

8. Find the area of the triangle whose vertices  are A(3, -1, 2) B(I, -1, -3) and C(4, -3,1)

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-c

9. Forces $2\overset { \wedge }{ i } +7\overset { \wedge }{ j }$$2\overset { \wedge }{ i } -5\overset { \wedge }{ j } +6\overset { \wedge }{ k }$$-\overset { \wedge }{ i } +2\overset { \wedge }{ j } -\overset { \wedge }{ k }$ act at a point P whose position vector is $4\overset { \wedge }{ i } -3\overset { \wedge }{ j } -2\overset { \wedge }{ k }$.Find  the vector moment of the resultant of these forces acting at P about this point Q whose position vector is $6\overset { \wedge }{ i } +\overset { \wedge }{ j } -3\overset { \wedge }{ k }$

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3

10. Find the Cartesian equation of a.line passing through the pointsA(2, -1, 3) and B(4, 2, 1)

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-1

11. Find the parametric form of vector equation of a line passing through a point (2, -1, 3) and parallel to line ${ \overset { \rightarrow }{ r } }=\left( \overset { \wedge }{ i } +\overset { \wedge }{ j } \right) +t\left( 2\overset { \wedge }{ i } +\overset { \wedge }{ j } -2\overset { \wedge }{ k } \right)$

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p=-1

12. Find the parametric form of vector equation of the plane passing through the point (1, -1, 2) having 2, 3, 3.as direction ratios of normal to the plane.

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2

13. If the planes ${ \overset { \rightarrow }{ r } }.\left( \overset { \wedge }{ i } +2\overset { \wedge }{ j } +3\overset { \wedge }{ k } \right) =7$=and ${ \overset { \rightarrow }{ r } }.\left( \lambda \overset { \wedge }{ i } +2\overset { \wedge }{ j } -7\overset { \wedge }{ k } \right) =26$ are perpendicular. Find the value of λ.

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∆=4

14. Flnd the equation of the plane containing the line of intersection of the planes x + y + Z - 6 = 0 and
2x + 3y + 4z + 5 = 0 and passing through the point (1, 1, 1)

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x = -1 is one root

15. Let $\overset { \rightarrow }{ a } ,\overset { \rightarrow }{ b } ,\overset { \rightarrow }{ c }$ be unit vectors such $\overset { \rightarrow }{ a } .\overset { \rightarrow }{ b } =\overset { \rightarrow }{ a } .\overset { \rightarrow }{ c } =0$ and the angle between $\overset { \rightarrow }{ b }$ and $\overset { \rightarrow }{ c }$ is $\frac { \pi }{ 6 }$Prove that $\overset { \rightarrow }{ a } =\pm 2\left( \overset { \rightarrow }{ b } \times \overset { \rightarrow }{ c } \right)$

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Type I even degree reciprocal equation