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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.

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

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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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