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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.

Let $straight a with rightwards arrow on top space equals space straight i with hat on top space minus space 2 space straight j with hat on top space minus space 3 space straight k with hat on top$ be the position vector of A and $straight i with hat on top space minus space straight j with hat on top space plus space 4 space straight k with hat on top comma space space space straight a with rightwards arrow on top space equals space 2 space straight i with hat on top space plus space straight j with hat on top space plus space 2 space straight k with hat on top$ be position vectors of B and C.
$therefore space space space space space space straight b with rightwards arrow on top space equals space BC with rightwards arrow on top space equals space straight P. straight V. space of space straight C space minus space straight P. straight V. space of space straight B space space space space space space space space space space space space space space equals space left parenthesis 2 space straight i with hat on top space plus space straight j with hat on top space plus space 2 space straight k with hat on top right parenthesis space minus space open parentheses straight i with hat on top space minus space straight j with ¨ on top space plus space 4 space straight k with ¨ on top close parentheses space equals space straight i with hat on top space plus space 2 space straight j with hat on top space minus space 2 space straight k with hat on top$
The vector equation of line through $straight A space open parentheses thin space straight a with rightwards arrow on top close parentheses space$ and parallel to vector $straight b with rightwards arrow on top$ is
$straight r with rightwards arrow on top space equals space straight a with rightwards arrow on top space plus space straight lambda space straight b with rightwards arrow on top$
or $straight r with rightwards arrow on top space equals space open parentheses straight i with rightwards arrow on top space minus space 2 space straight j with hat on top space minus space 3 space straight k with hat on top close parentheses space plus straight lambda space open parentheses straight i with hat on top space plus space 2 space stack straight j space with hat on top space minus space 2 space straight k with hat on top close parentheses$
Now taking $straight r with rightwards arrow on top space equals space straight x space straight i with hat on top space plus space straight y space straight j with hat on top space plus space straight z space straight k with hat on top$ , the equation of line is
$straight x space straight i with hat on top space plus space straight y space straight j with hat on top space plus space straight z space straight k with hat on top space equals space open parentheses straight i with hat on top space minus 2 space straight j with hat on top space minus space 3 space straight k with hat on top close parentheses space plus space straight lambda space open parentheses straight i with hat on top plus 2 space straight j with hat on top space minus space 2 space straight k with hat on top close parentheses$
or $straight x space straight i with hat on top space plus space straight y space straight j with hat on top space plus space straight z space straight k with hat on top space equals space left parenthesis straight lambda plus 1 right parenthesis space straight i with hat on top space plus space left parenthesis 2 space straight lambda space minus space 2 right parenthesis space straight j with hat on top space minus space left parenthesis 2 straight lambda plus 3 right parenthesis space straight k with hat on top$
Comparing the coefficients of $straight i with hat on top comma space straight j with hat on top comma space straight k with hat on top$ we get, x = λ + 1, y = 2 λ – 2, z = – (2 λ + 3)
or $fraction numerator straight x minus 1 over denominator 1 end fraction space equals space fraction numerator straight y plus 2 over denominator 2 end fraction space equals space fraction numerator straight z plus 3 over denominator negative 2 end fraction space left parenthesis equals space straight lambda right parenthesis$ which is cartesian equation of line.