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Applications of Vector Algebra - Two Marks Study Materials

12th Standard EM

    Reg.No. :
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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 \(\overset { \rightarrow }{ c } \)

    ()

    -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.

    ()

    b

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

    ()

    -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 \(6\overset { \wedge }{ i } +\overset { \wedge }{ j } -3\overset { \wedge }{ k } \)

    ()

    3

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

    ()

    -1

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

    ()

    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.

    ()

    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.

    ()

    ∆=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)

    ()

    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 \(\overset { \rightarrow }{ a } =\pm 2\left( \overset { \rightarrow }{ b } \times \overset { \rightarrow }{ c } \right) \)

    ()

    Type I even degree reciprocal equation

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