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Show that [veca+vecbvecb+veccvecc+veca] ...

Show that `[veca+vecbvecb+veccvecc+veca] = 2[vecavecbvecc]`

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Prove that [vecaxxvecbvecbxxveccveccxxveca] = [vecavecbvecc]^2

Show that veca, vecb and vec c are coplanar if veca+vecb,vecb+vecc and vecc+veca are coplanar.

For any three vectors veca,vecb,vecc show that [(veca-vecb) (vecc-veca) (vecb-vecc)] = 0

Show that (veca-vecb)xx(veca+vecb) = 2(vecaxxvecb) . Interpret this result geometrically.

For veca = hati+hatj, vecb = -hati+2hatk, vecc = hatj+hatk , obtain vecaxx(vecbxxvecc) and also verify the formula vecaxx(vecbxxvecc) = (veca.vecc)vecb-(veca.vecb)vecc .

If veca, vecb, vecc are mutually perpendicular vectors of equal magnitude, show that veca+vecb+vecc is equally inclined to veca.vecb.vecc .

If a and b are perpendicular vectors show that (veca+vecb)^2 = (veca-vecb)^2 . [(veca+vecb)^2 means (veca+vecb).(veca+vecb) , so does (veca-vecb)^2 .]

If veca,vecb,vecc are such that veca.vecb = veca.vecc then show that veca = vec0 or vecb = vecc or veca is perpendicular to vecb.vecc .

|veca| = |vecb| = veca = vecb .

Prove that the four points with position vectors 2veca+3vecb-vecc, veca-2vecb+3vecc, 3veca+4vecb-2vecc and veca-6vecb+6vecc are coplanar.

SHARAM PUBLICATION-THREE DIMENSIONAL GEOMETRY-EXAMPLE
  1. The position vectors of two points A and B are 3hati+hatj+2hatk and ha...

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  2. Find the area of the triangle ABC with vertices A(1,2,4), B(3,1,-2) an...

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  3. Show that [veca+vecbvecb+veccvecc+veca] = 2[vecavecbvecc]

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  4. If the sum of two unit vectors is a unit vector, show that the magnitu...

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  5. Find the shortest distance between the following lines vecr = 2hat(i) ...

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  6. Find the shortest distance between the lines  (x-8)/(3) = (y+9)/(-16) ...

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  7. Find the distance of the point (-1,-5,-10) from the point of intersect...

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  8. The projection of a line segment overline(OP), through origin O, on th...

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  9. passing through the point (-1,3,2) perpendicular to the planes x+2y+...

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  10. Prove that the lines (x+4)/3=(y+6)/5=(z-1)/(-2) and 3x-2y+z+5=0=2x+3y...

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  11. Find the locus of points which are equidistant from the points (1,2,3)...

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  12. Find the equation of the plane through the points (1, 2, -3), (2,3, -4...

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  13. Find the perpendicular distance of the point (-1,3,9) from the line (x...

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  14. Prove that the measure of the angle between two main diagonals of a cu...

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  15. If P (1,y,z) lies on the line through (3,2,-1) and (-4,6,3) find y & z...

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  16. Find the equation of the plane passing through the foot of the perpend...

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  17. Find the equation of the plane passing through the points (2, 1, -1) a...

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  18. Find the points of intersection of the line (x-1)/1=(y+2)/ 3 =(z-1)/(...

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  19. Find the ratio in which the line segment through (1,3,-1) and (2,6,-2)...

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  20. Find the co-ordinates of the point where the perpendicular from the or...

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