Maths Book back answers and solution for Exercise questions - Mathematics : Scalar Product and Vector Product

EXERCISE 6.1

1. Prove by vector method that if a line is drawn from the centre of a circle to the midpoint of a chord, then the line is perpendicular to the chord.

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

3. Prove by vector method that an angle in a semi-circle is a right angle.

4. Prove by vector method that the diagonals of a rhombus bisect each other at right angles.

5. Using vector method, prove that if the diagonals of a parallelogram are equal, then it is a rectangle.

6. Prove by vector method that the area of the quadrilateral *ABCD *having diagonals *AC *and BD is .

7. Prove by vector method that the parallelograms on the same base and between the same parallels are equal in area.

8. If *G *is the centroid of a Î”*ABC *, prove that (area of Î”*GAB*) = (area of Î”*GBC*) = (area of Î”*GCA*) = 1/3 (area of Î”*ABC*)

9. Using vector method, prove that cos(*Î±** *âˆ’ *Î²** *) = cos*Î±** *cos *Î²** *+ sin *Î±** *sin *Î²** *.

10. Prove by vector method that sin(*Î±** *+ *Î²** *) = sin *Î±** *cos *Î²** *+ cos*Î±** *sin *Î²** *.

11. A particle acted on by constant forces 8*i*Ë† + 2 Ë†*j *âˆ’ 6*k*Ë† and 6*i*Ë† + 2 Ë†*j *âˆ’ 2*k*Ë† is displaced from the point (1, 2, 3) to the point (5, 4,1) . Find the total work done by the forces.

12.Forces of magnitudes 5âˆš2 and 10âˆš2 units acting in the directions 3*i*Ë† + 4 Ë†*j *+ 5*k*Ë† and 10*i*Ë† + 6 Ë†*j *âˆ’ 8*k*Ë† , respectively, act on a particle which is displaced from the point with position vector 4*i*Ë† âˆ’ 3 Ë†*j *âˆ’ 2*k*Ë† to the point with position vector 6*i*Ë† + Ë†*j *âˆ’ 3*k*Ë† .Find the work done by the forces.

13. Find the magnitude and direction cosines of the torque of a force represented by 3*i*Ë† + 4 Ë†*j *âˆ’ 5*k*Ë† about the point with position vector 2*i*Ë† âˆ’ 3 Ë†*j *+4*k*Ë† acting through a point whose position vector is 4*i*Ë† + 2 Ë†*j *âˆ’ 3*k*Ë† .

14. Find the torque of the resultant of the three forces represented by âˆ’3*i*Ë† + 6 Ë†*j *âˆ’ 3*k*Ë† , 4*i*Ë† âˆ’10 Ë†*j *+12*k*Ë† and 4*i*Ë† + 7 Ë†*j *acting at the point with position vector 8*i*Ë† âˆ’ 6 Ë†*j *âˆ’ 4*k*Ë† , about the point with position vector 18*i*Ë† + 3 Ë†*j *âˆ’ 9*k*Ë† .

Answers:

11. 80 units

12. 69 units

13. âˆš179 , 3/âˆš179 , 11/âˆš179, âˆ’ , 7/âˆš179

14. -96Ë†*i* + 115Ë†*j* + 15Ë†*k*

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12th Mathematics : UNIT 6 : Applications of Vector Algebra : Exercise 6.1: Scalar Product and Vector Product | Problem Questions with Answer, Solution

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