The cartesian equations of a line are 6 x – 2 = – 3 y + 1 =

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Three Dimensional Geometry

Long Answer Type

61. Find the vector equation of a line passing through a point with position vector

2 straight i with hat on top space minus space straight j with hat on top space plus space straight k with hat on top

and parallel to the line joining the points with position vectors

negative straight i with hat on top space plus space 4 space straight j with hat on top space plus space straight k with hat on top

and

straight i with hat on top space plus space 2 space straight j with hat on top space plus space 2 space straight k with hat on top

. Also, find the cartesian equivalent of this equations.

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Short Answer Type

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62. The cartesian equations of a line are 6 x – 2 = – 3 y + 1 = 2 z = 2. Find the direction ratios of the line and write down the vector equation of the line through (2, – 1, –1) which is parallel to the given line.

The equation of the line are 6 x – 2 = 3 y + 1 = 2 z – 2
or

6 space open parentheses straight x minus 1 third close parentheses space equals space 3 open parentheses straight y plus 1 third close parentheses space equals space 2 left parenthesis straight z minus 1 right parenthesis

or

fraction numerator straight x minus begin display style 1 third end style over denominator 1 end fraction space equals space fraction numerator straight y minus open parentheses negative begin display style 1 third end style close parentheses over denominator 2 end fraction space equals space fraction numerator straight z minus 1 over denominator 3 end fraction

∴ direction-ratios of the line are 1, 2, 3.

therefore

straight m with rightwards arrow on top space equals space straight i with hat on top space plus space 2 space straight j with hat on top space plus space 3 space straight k with hat on top

Required line passes through (2, – 1, – 1)

therefore space space space space straight a with rightwards arrow on top space equals space 2 space straight i with hat on top space minus space straight j with hat on top space minus space straight k with hat on top

Let

straight r with rightwards arrow on top

be the position vector of any point on the line.
∴ the vector equation of required line is

straight r with rightwards arrow on top space equals space straight a with rightwards arrow on top space plus space straight lambda straight m with rightwards arrow on top space space space or space space space straight r with rightwards arrow on top space equals space left parenthesis 2 space straight i with hat on top space minus space straight j with hat on top space minus space straight k with hat on top right parenthesis space space plus space straight lambda space open parentheses straight i with hat on top space plus space 2 space straight j with hat on top space plus space 3 space straight k with hat on top close parentheses

where λ is a parameter.

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63. The points A (4, 5, 10), B (2, 3, 4) and C (1, 2, – 1) are three vertices of a parallelogram ABCD. Find vector and cartesian equations for the sides AB and BC and find the coordinates of D.

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Long Answer Type

64. A line passes through the point with position vector

2 space straight i with hat on top space minus space straight j with hat on top space plus space 4 space straight k with hat on top

and is in direction

straight i with hat on top space plus space straight j with hat on top space minus space 2 space straight k with hat on top.

Find equations for the line in vector and in cartesian form .

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65. Find the vector equation of a line passing through a point with position vector

straight i with hat on top space minus space 2 space straight j with hat on top space minus space 3 space straight k with hat on top

and parallel to the line joining the points with position vectors

straight i with hat on top space minus space straight j with hat on top space plus space 4 space straight k with hat on top

and

2 space straight i with hat on top space plus space straight j with hat on top space plus space space 2 space straight k with hat on top.

Also, find the cartesian equivalent of this equation.

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Short Answer Type

66. Find the angle between pair of lines:

straight r with rightwards arrow on top space equals space 3 space straight i with hat on top space plus space 2 space straight j with hat on top space minus space 4 space straight k with hat on top space plus space straight lambda space left parenthesis straight i with hat on top space plus space 2 space straight j with hat on top space plus space 2 space straight k with hat on top right parenthesis

and

straight r with rightwards arrow on top space equals space 5 space straight i with hat on top space minus space 2 space straight k with hat on top space plus space straight mu space left parenthesis 3 space straight i with hat on top space plus space 2 space straight j with hat on top space plus space 6 space straight k with hat on top right parenthesis

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67. Find the angle between the pair of lines:

straight r with rightwards arrow on top space equals space 2 space straight i with hat on top space minus space 5 space straight j with hat on top space plus space straight k with hat on top space plus space straight lambda space left parenthesis 3 space straight i with hat on top space plus space 2 space straight j with hat on top space plus space 6 space straight k with hat on top right parenthesis

and

straight r with rightwards arrow on top space equals space 7 space straight i with hat on top space minus space 6 space straight k with hat on top space plus space straight mu space left parenthesis straight i with hat on top space plus space 2 space straight j with hat on top space plus space 2 space straight k with hat on top right parenthesis

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68. Find the angle between the pair of lines:

straight r with rightwards arrow on top space equals space 3 space straight i with hat on top space plus space straight j with hat on top space minus space 2 space straight k with hat on top space plus space straight lambda space left parenthesis straight i with hat on top space minus space straight j with hat on top space minus space 2 space straight k with hat on top right parenthesis

and

straight r with rightwards arrow on top space equals space 2 space straight i with hat on top space minus space straight j with hat on top space minus space 5 space straight k with hat on top space plus space straight mu space left parenthesis 3 space straight i with hat on top space minus space 5 space straight j with hat on top space minus space 4 space straight k with hat on top right parenthesis

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69. Find the angle between the pair of lines with direction ratios 2, 2, 1 and 4, 1, 8.

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70. Find the angle between the pair of lines with direction ratios 2, 6, 3 and 1, 2, 2.

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