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Vector Algebra — Class 12 solved problems

22 problems from this chapter, each solved step by step.

  1. Find the angle between the vectors a = 2i + 3j + k and b = i - 2j + 3k.

    The angle between the vectors is cos^(-1)(-(1)/(14)).

  2. Let a = i + j + k, b = 2 i + 2 j + k and d = a × b. If c is a vector such that a · c = | c|, | c - 2 a|² = 8 and the angle between d and c is (π)/(4), then |10

    6

  3. Consider two vectors u=3i-j and v=2i+j-, > 0. The angle between them is given by cos^(-1) ((√5)/(2√7)). Let v=v_1+v_2 where v_1 is parallel to u and v_2 is perp

    14

  4. If the components of a=α i+β j+ k along and perpendicular to b= i+ j- k respectively, are (16)/(11)(3 i+ j- k) and (1)/(11)(-4 i-5 j-17 k), then α²+β²+ ² is equ

    26

  5. Let a = i + 2 j + k and b = 2 i + j - k. Let c be a unit vector in the plane of the vectors a and b and be perpendicular to a. Then such a vector c is:

    c = ± (1)/(√2) (- i + k)

  6. Let the point A divide the line segment joining the points P(-1,-1,2) and Q(5,5,10) internally in the ratio r:1 (r>0). If O is the origin and ( OQ · OA - (1)/(5

    7

  7. Let a be a unit vector perpendicular to the vectors b= i-2 j+3 k and c=2 i+3 j- k, and makes an angle of cos^(-1)(-(1)/(3)) with the vector i+ j+ k. If a makes

    The value of α is -√6.

  8. Let a = i + 2 j + 3 k, b = 3 i + j - k and c be three vectors such that c is coplanar with a and b. If the vector c is perpendicular to b and a · c = 5, then |

    (√66)/(6)

  9. Let the three sides of a triangle ABC be given by the vectors 2 i- j+ k, i-3 j-5 k and 3 i-4 j-4 k. Let G be the centroid of the triangle ABC. Then 6 (| AG|²+|

    164

  10. Let c be the projection vector of b = i + 4 k, > 0, on the vector a = i + 2 j + 2 k. If | a + c| = 7, then the area of the parallelogram formed by the vectors b

    16

  11. Let the angle θ, 0 < θ < (π)/(2), be the angle between two unit vectors a and b, and θ = sin^(-1)((√65)/(9)). If the vector c = 3 a + 6 b + 9( a × b), then the

    29

  12. If a is a nonzero vector such that its projections on the vectors 2 i- j+2 k, i+2 j-2 k, and k are equal, then a unit vector along a is:

    ± (1)/(√155)(7 i + 9 j + 5 k)

  13. Let a = 3 i - j + 2 k, b = a × ( i - 2 k), and c = b × k. Then the projection of c - 2 j on a is:

    -(24)/(√14)

  14. Let the arc AC of a circle subtend a right angle at the centre O. If the point B on the arc AC, divides the arc AC such that (length of arc AB)/(length of arc B

    2 - √3

  15. Let a=2 - +3 k, b=3 -5 + k and c be a vector such that a × c= c × b and ( a+ c) · ( b+ c)=168. Then the maximum value of | c|² is:

    308

  16. Let a and b be two unit vectors such that the angle between them is (π)/(3). If a + 2 b and 3 a - b are perpendicular to each other, then the number of values o

    0

  17. Let the vectors a and b be of the same magnitude such that ( a + b + a - b)/( a + b - a - b) = √2 + 1. Then ( a + b ²)/( a ²) is:

    2 + √2

  18. Let A, B, C be three points in xy -plane, whose position vectors are given by √3 i+ j, i+√3 j and a i+(1-a) j respectively with respect to the origin O. If the

    1

  19. Let the position vectors of three vertices of a triangle be 4 p + q - 3 r, -5 p + q + 2 r, and 2 p - q + 2 r. If the position vectors of the orthocenter and the

    2

  20. Let a= i+2 j+ k, b=3 i-3 j+3 k, c=2 i- j+2 k and d be a vector such that b× d= c× d and a· d=4. Then | a× d|² is equal to:

    128

  21. Let a = 2 i - 3 j + k, b = 3 i + 2 j + 5 k and a vector c be such that ( a - c) × b = -18 i - 3 j + 12 k and a · c = 3. If b × c = d, then | a · d| is equal to:

    15

  22. Let the position vectors of the vertices A, B and C of a tetrahedron ABCD be i + 2 j + k, i + 3 j - 2 k and 2 i + j - k respectively. The altitude from the vert

    The position vector of E is (13)/(9) i + 2 j - (11)/(9) k.

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