Find the angle between pair of lines:

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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

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.

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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

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

The equations of two lines are

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

therefore

stack straight b subscript 1 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 2 space straight k with hat on top comma space space space space space stack straight b subscript 2 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 plus space 6 space straight k with hat on top

Let θ be the angle between the lines.

therefore space space space space space space cos space straight theta space equals space fraction numerator stack straight b subscript 1 with rightwards arrow on top space. space stack straight b subscript 2 with rightwards arrow on top over denominator open vertical bar stack straight b subscript 1 with rightwards arrow on top close vertical bar space open vertical bar stack straight b subscript 2 with rightwards arrow on top close vertical bar end fraction space space space space space space space space space space space space space space space space space space space space space equals space fraction numerator left parenthesis 1 right parenthesis thin space left parenthesis 3 right parenthesis space plus space left parenthesis 2 right parenthesis thin space left parenthesis 2 right parenthesis space plus space left parenthesis 2 right parenthesis thin space left parenthesis 6 right parenthesis over denominator square root of 1 plus 4 plus 4 end root space square root of 9 plus 4 plus 36 end root end fraction space equals space fraction numerator 3 plus 4 plus 12 over denominator square root of 9 space square root of 49 end fraction space equals space fraction numerator 19 over denominator 3 cross times 7 end fraction space equals space 19 over 21 therefore space space space space space space space space straight theta space equals space cos space to the power of negative 1 end exponent space open parentheses 19 over 21 close parentheses.

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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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