Find the angle between the pair of lines with direction ratios 2

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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

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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.

The direction ratios of two lines are 2, 6, 3 and 1, 2, 2.
Let θ be the angle between the lines

therefore space space space space space cos space straight theta space equals space fraction numerator left parenthesis 2 right parenthesis thin space left parenthesis 1 right parenthesis space plus space left parenthesis 6 right parenthesis thin space left parenthesis 2 right parenthesis space plus space left parenthesis 3 right parenthesis thin space left parenthesis thin space 2 right parenthesis over denominator square root of 4 plus 36 plus 9 end root space square root of 1 plus 4 plus 4 end root end fraction space equals space fraction numerator 2 plus 12 plus 6 over denominator square root of 49 space square root of 9 end fraction space equals space fraction numerator 20 over denominator 7 cross times 3 end fraction space equals space 20 over 21 therefore space space space space space space space space space space straight theta space equals space cos to the power of negative 1 end exponent open parentheses 20 over 21 close parentheses

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