Find the angle between the pair of lines: and

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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

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

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 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 stack straight k right parenthesis with hat on top

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

therefore space space space space space space space straight b with rightwards arrow on top space equals space 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 comma space space stack straight b apostrophe with rightwards arrow on top space equals space 3 space straight i with hat on top space minus space 5 space straight j with hat on top minus space 4 space straight k with hat on top

Let θ be angle between the lines.

therefore space space space space space space cos space straight theta space equals space fraction numerator straight b with rightwards arrow on top. space stack straight b apostrophe with rightwards arrow on top over denominator open vertical bar straight b with rightwards arrow on top close vertical bar space open vertical bar straight b with rightwards arrow on top apostrophe 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 negative 1 right parenthesis thin space left parenthesis negative 5 right parenthesis space plus space left parenthesis negative 2 right parenthesis thin space left parenthesis negative 4 right parenthesis over denominator square root of 1 plus 1 plus 4 end root space square root of 9 plus 25 plus 16 end root end fraction space equals space fraction numerator 3 plus 5 plus 8 over denominator square root of 6 space square root of 50 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 16 over denominator square root of 6. space 5 square root of 2 end fraction space equals space 16 over 5 cross times fraction numerator 1 over denominator square root of 12 end fraction space equals space 16 over 5 cross times fraction numerator 1 over denominator 2 square root of 3 end fraction cross times fraction numerator square root of 3 over denominator square root of 3 end fraction space equals fraction numerator 8 square root of 3 over denominator 15 end fraction therefore space space space space space space space space straight theta space equals space cos to the power of negative 1 end exponent open parentheses fraction numerator 8 square root of 3 over denominator 15 end fraction close parentheses.

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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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