VECTORS
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1. If a = 2 î +3 ˆj +4 k̂ and a = 2 î -2 ˆj +3 k̂ , find (a) a . b (b) find the angle between â & b̂
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(c) projection of a on b̂ (d) projection of b̂ on a (e) Angle between a and â + b̂ (f)
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unit vector in the direction a - b̂ .
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2. If â + b + ĉ = 0 | â | = 3 | b | = 4 | ĉ | = 7, find the angles between a & b , b & ĉ and
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a & ĉ
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3. If a . b = 6, | a | = 3 , | b | = 4 find the angle between | a | and | b |
4. Prove that the diagonals of a rhombus are perpendicular to each other.
5. If the diagonals of a rhombus are perpendicular to each other then prove that it is a
rhombus.
6. Prove that the diagonals of a rectangle are equal.
7. If the diagonals of a parallelogram are equal then prove that it is a rectangle.
8. Prove that the quadrilateral formed by joining the mid points of a rectangle is a rhombus.
9. Prove that the mid point of the hypotenuse of a right triangle is the circumcentre .
10. Prove that the in an isosceles triangle, the median is perpendicular the base.
11. If the median of a triangle is perpendicular to the base then prove that the triangle is
isosceles.
12. If AD be a median of a triangle ABC, prove that AB2 + AC2 = 2(AD2 + BD2)
13. Prove that the angle in a semi circle is right angled.
14. In an isosceles. Prove that the median corresponding to equal sides are equal.
15. If two medians of a triangle are equal then prove that it is an isosceles.
16. If the sum of the two unit vectors is a unit vector then prove that the modulus of their
difference is 3
17. If the difference of the two unit vectors is a unit vector then prove that the modulus of
their sum is 3
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18. Prove that | a + b |  | â | + | b |
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19. Prove that | a - b |  | a | - | b |
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20. Prove that Cauchy – Schwartz inequality ( a . b )2  a2 b2
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21. If  is the angle between two vectors a & b prove that | a + b | = 2cos(/2), and | a - b |
= 2sin(/2)
22. Prove that altitudes of a triangle are concurrent.
23. Prove that the perpendicular bisector of the sides of a triangle are concurrent
24. State and prove the cosine law of triangle.
25. State and prove the projection formula of triangles.
26. Prove that cos(A+B) = cosAcosB – sinAsinB
27. Prove that cos(A-B) = cosAcosB + sinAsinB
28. Prove that the diagonals of an isosceles trapezium are equal.
29. If the diagonals of a trapezium are equal prove that the it is isosceles.
30. If two pair of opposite edges of a tetrahedron are perpendicular to each other, prove that
the third pair is also perpendicular. Also prove that the sum of the squares of the opposite
edges is a constant.
Vector product
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If a = 3 î + ˆj + 2 k̂ , b = 2 î -2 ˆj + 4 k̂ , find
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(a) a x b (b) a unit vector perpendicular to a & b (c) a vector of magnitude 10 and
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perpendicular to a & b (d) sine and cosine of angles between a & b (e) Area of the
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parallelogram whose sides are a & b .
If A(1,1,2) , B( 2,3,4), C(3,-1,5) find (a) area of triangle ABC, (b) angles of triangle ABC,
(c) unit vector perpendicular to triangle ABC (d) vector of magnitude 5 perpendicular to
ABC.
3.
The diagonals of a parallelogram are
(a) d1 = 2 î + ˆj + 3 k̂ , and d2 = 3 î -2 ˆj -3 k̂ (b) d1 = 3 î + 2 ˆj + k̂ , and d2 = 2 î +4 ˆj -5 k̂
find the are aof the parallelogram and unit vector perpendicular to the parallelogram.
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5.
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If a = 2 î + 3 ˆj + 6 k̂ , b = 3 î -6 ˆj + 2 k̂ and c = 6 î +2 ˆj -3 k̂ prove that a x b = 7 c
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If â , a & ĉ are the position vectors of the vertices of a triangle prove that the area of the
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triangle ABC = ½ ( a x a + a x ĉ + ĉ x a )
6.
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( a x b )2 + ( a . b )2 = ( a . a )( b . a )
7.
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Prove that a x ( b + c ) + b x( c + a ) + c x( a + b ) = 0
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Stae and prove sine law of triangles
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Prove that the area of quadrilateral ABCD ½ ACxBD
10. Prove that sin(A+B) = sinACosB + cosAsinB
11. Prove that sin(A-B) = sinACosB – cosAsinB
12. A,B,C and Dare the 4 points in space prove that |ABxCD + BC x AD + CA x BD| = 4times
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13. If a x b = c x d show that  0 show that a + c = k b were k is a scalar
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14. If a = 3 î + x ˆj - k̂ , is perpendicular to b = 2 î + ˆj + y k̂ and a = b .
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15. If a x b = c x d and a x c = b x d prove that a - d parallel to b - c .
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16. If a = î - ˆj & b = - î +2 k̂ prove that ( a + b ).( a -2 b ) = -9.
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17. Show that ( a - b )x( a + b ) = 2( a x b )and interpret it geometrically.
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18. If a , b c are three vectors a x b = c , b x c = a , prove that the points are orthogonal in
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pairs and | b |=1 and a = c .
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19. If a . b = a . c and a x b = a x c , a  0, prove that b = c
20. Find the work done by the force F = 2 î + ˆj + k̂ in displacing a particle from A(3,1,2) to B(1,4,6)
21. Find the torque of the force F = 3 î + 2 ˆj - k̂ acting at A(-1,0,7) and rotating the body about
O(7,6,5).