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Sequences And Series

Question
CBSEENMA11015469
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The angle between the lines whose direction cosines satisfy the equations l +m+n=0 and l2 = m2+n2 is

  • π/3

  • π/4

  • π/6

  • π/2

Solution

A.

π/3

We know that angle between two lines is 
cos space straight theta space equals space fraction numerator straight a subscript 1 straight a subscript 2 space plus straight b subscript 1 straight b subscript 2 space plus straight c subscript 1 straight c subscript 2 over denominator square root of straight a subscript 1 superscript 2 plus straight b subscript 1 superscript 2 plus straight c subscript 1 superscript 2 end root square root of straight a subscript 2 superscript 2 plus straight b subscript 2 superscript 2 plus straight c subscript 2 superscript 2 end root end fraction
l +m +n= 0
⇒ l = - (m+n)
⇒ (m+n)2 = l2
⇒ m2 +n2 +2mn = m2 +n2
[∵ l2 = m2 +n2, given]
⇒ 2mn = 0
when m = 0 ⇒ l =-n
Hence, (l, m, n) is (1,0-1)
When n =0, then l =-m
Hence, (l,m,n) is (1,0-1)
Cos space straight theta space equals space fraction numerator 1 plus 0 plus 0 over denominator square root of 2 space end root space straight x square root of 2 end fraction space equals space 1 half rightwards double arrow straight theta space equals space straight pi over 3

Some More Questions From Sequences and Series Chapter

Find the octant in which the following points lie:
A (-3,1, 2), B (-3, 1, -2), C (-3, -1,-2), D (3, -1, -2) and E (3,1, 2).

A point lies on the x-axis at a distance of 5 units towards OX'. What are its co-ordinates?

A point lies on z-axis at a distance of 3 units in the direction OZ. Write its co-ordinates.

A point lies on y-axis and is at a distance of 4 units in the direction OY'. Write its co-ordinates.

A point lies on XY-plane with distances 4 units along OX and 3 units along OY. Write its co-ordinates.

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A point lies on ZX-plane with distances 3 units along OX' and 5 units along OZ'. Write its co-ordinates.

Find the perpendicular distance of point A (-3, 2, -4) from:
(a) ZX-plane      (b)XY-plane         (c) YZ-plane.

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