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If `(l_1, m_1, n_1), (l_2, m_2, n_2)" and "(l_3, m_3, n_3)` are the direction cosines of three mutually perpendicular lines, prove that the line whose direction cosines are proportional to `(l_1+l_2+l_3,m_1+m_2+m_3,n_1+n_2+n_3)` makes equal angles with them,

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The correct Answer is:
`therefore alpha=beta=gamma`
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If r_(1),r_(2) and r_(3) are the position vectors of three collinear points and scalars l and m exists such that r_(3)=lr_(1)+mr_(2) , then show that l+m=1.

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KUMAR PRAKASHAN-THREE DIMENSIONAL GEOMETRY-SOLUTION OF NCERT EXAMPLAR PROBLEMS (Short Answer Type Question)
  1. vec(AB)=3bari-barj+bark and vec(CD)= -3bari+2barj+4bark are two vector...

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  2. Show that the straight lines whose direction cosines are given by 2l +...

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  3. If (l1, m1, n1), (l2, m2, n2)" and "(l3, m3, n3) are the direction cos...

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  4. Distance of the point (alpha,beta,gamma) from Y- axis is …........

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  5. If the directions cosines of a line are k, k and k then ..........

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  6. The distance of the plane barr.(2/7i+3/7j-6/7k)=1 from the origin is …...

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  7. The sine of the angle between the straight line (x-2)/(3)=(y-3)/(4)=(z...

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  8. The reflection of the point (alpha,beta,gamma) in the xy - plane is .....

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  9. The area of the quadrilateral ABCD, where A(0, 4, 1), B(2, 3, -1), C(4...

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  10. The locus represented by xy + yz = 0 is .........

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  11. The plane 2x – 3y + 6z - 11 = 0 makes an angle sin^(-1)alpha with X- a...

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  12. A plane passes through the points (2, 0, 0) (0,3,0) and (0, 0, 4). The...

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  13. The direction cosines of the vector 2bari + 2barj - bark are ............

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  14. The cartesian equation of a line is (x-5)/(3)=(y+4)/(7)=(z-6)/(2).Writ...

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  15. The vector equation of the line through the points (3, 4, -7) and (1, ...

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  16. The cartesian equation of the plane barr.(bari+barj-bark)=2 is...........

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  17. The unit vector normal to the plane x + 2y + 3z - 6 = 0 is (1)/(sqrt(1...

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  18. The intercepts made by the plane 2x - 3y + 5z + 4 = 0 on the co-ordina...

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  19. The angle between the line vecr=(5hati-hatj-4hatk)+lambda(2hati-hatj+h...

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  20. The angle between the planes barr.(2bari-3barj+bark)=1 and barr.(i-j)=...

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