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Three Dimensional Geometry

Short Answer Type

41. The equation of a line is

fraction numerator 2 straight x minus 5 over denominator 4 end fraction space equals space fraction numerator straight y plus 4 over denominator 3 end fraction space equals fraction numerator 6 minus straight z over denominator 6 end fraction

. Find the direction cosines of a line parallel to this line.

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Long Answer Type

42. The cartesian equations of a line are 3 x + 1 = 6 y – 2 = 1 – z. Find the fixed point through which it passes, its direction ratios and also its vector equation.

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Short Answer Type

43. Find the equation of the line which passes through the point (3, 4, 5) and is parallel to the vector

2 straight i with hat on top plus 2 straight j with hat on top minus 3 straight k with hat on top.

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44. Find the vector and Cartesian equations of the line through the point (5, 2, – 4) and which is parallel to the vector $3 straight i with overparenthesis on top space plus space 2 straight j with overparenthesis on top space minus space 8 space straight k with overparenthesis on top.$

Since the line passes through the point (5, 2, 4) and is parallel to the vector

3 straight i with hat on top space plus space 2 straight j with hat on top space minus space 8 straight k with hat on top.

therefore space space space space straight a with rightwards arrow on top space equals space 5 straight i with hat on top space plus space 2 straight j with hat on top space minus space 4 straight k with hat on top comma space space space space straight m with rightwards arrow on top space equals space 3 straight i with hat on top space plus space 2 straight j with hat on top space minus space 8 space straight k with hat on top

Let

straight r with rightwards arrow on top

be the position vector of any point on the line.
∴ the vector equation of line is

straight r with rightwards arrow on top space equals space straight a with rightwards arrow on top plus space straight lambda straight m with rightwards arrow on top

straight r with rightwards arrow on top space equals space left parenthesis 5 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 right parenthesis space plus space straight lambda left parenthesis 3 straight i with hat on top space plus space 2 straight j with hat on top space minus space 8 straight k with hat on top right parenthesis

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 straight i with hat on top space plus space straight y straight j with hat on top space plus space straight z straight k with hat on top space equals space left parenthesis 5 plus 3 straight lambda right parenthesis space straight i with hat on top space plus space left parenthesis 2 plus 2 straight lambda right parenthesis space straight j with hat on top space plus space left parenthesis negative 4 minus 8 space straight lambda 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 comma space

we get,

straight x space equals 5 plus 3 straight lambda comma space space straight y space equals space 2 plus 2 straight lambda comma space space space straight z space equals space minus 4 minus 8 straight lambda

fraction numerator straight x minus 5 over denominator 3 end fraction space equals space fraction numerator straight y minus 2 over denominator 2 end fraction space equals space fraction numerator straight z plus 4 over denominator negative 8 end fraction space equals space left parenthesis straight lambda right parenthesis

which is cartesian equation of line.

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45.

Find the equation of the line which passes through the point (1, 2, 3) and is parallel to the vector $3 straight i with overparenthesis on top space plus space 2 straight j with overparenthesis on top space minus space 2 straight k with overparenthesis on top.$