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Three Dimensional Geometry

Long Answer Type

31. The projection of a line on the co-ordinate axes are 6, 2, 3. Find the length of the line and its direction cosines. Also, find the projection of the line segment joining (2, – 3, 1) to (4, 2, 3) on this line.

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32. A (– 1, 2, – 3), B (5, 0, – 6), C (0, 4, – 1) are three points. Show that the direction cosines of the bisectors of are proportional to 25, 8, 5 and -11,

AB space equals space square root of left parenthesis 5 plus 1 right parenthesis squared plus left parenthesis 0 minus 2 right parenthesis squared plus left parenthesis negative 6 plus 3 right parenthesis squared end root space space space space space space equals space square root of 36 plus 4 plus 9 end root space equals space square root of 49 space equals space 7 AC space equals space square root of left parenthesis 0 plus 1 right parenthesis squared plus left parenthesis 4 minus 2 right parenthesis squared plus left parenthesis negative 1 plus 3 right parenthesis squared end root space space space space space space space space equals space square root of 1 plus 4 plus 4 end root space equals space square root of 9 space equals space 3

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Short Answer Type

33. The cartesian equations of a line are

fraction numerator straight x minus 5 over denominator 3 end fraction space equals space fraction numerator straight y plus 4 over denominator 7 end fraction space equals fraction numerator straight z minus 6 over denominator 2 end fraction.

Find a vector equation for the line.

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34.

The Cartesian equation of a line is $fraction numerator straight x plus 3 over denominator 2 end fraction space equals space fraction numerator straight y minus 5 over denominator 4 end fraction space equals fraction numerator straight z plus 6 over denominator 2 end fraction.$ Find the vector equation for the line.

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35.

Find the cartesian equation of the line which passes through the point (-2, 4, -5) and is parallel to the line given by $fraction numerator straight x plus 3 over denominator 3 end fraction space equals space fraction numerator straight y minus 4 over denominator 5 end fraction space equals fraction numerator straight z plus 8 over denominator 6 end fraction$

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36.

Find the equations of a line which is parallel to the line $fraction numerator straight x minus 2 over denominator 3 end fraction space equals fraction numerator straight y plus 1 over denominator 1 end fraction space equals space fraction numerator straight z minus 7 over denominator 9 end fraction$ and which passes through the point (3, 0, 5).

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Long Answer Type

37.
For the cartesian and vector equation of a line which passes through the point (1, 2, 3) and is parallel to the line $fraction numerator negative straight x minus 2 over denominator 1 end fraction space equals space fraction numerator straight y plus 3 over denominator 7 end fraction space equals space fraction numerator 2 straight z minus 6 over denominator 3 end fraction.$

The given line is

fraction numerator negative straight x minus 2 over denominator 1 end fraction space equals space fraction numerator straight y plus 3 over denominator 7 end fraction space equals space fraction numerator 2 straight z minus 6 over denominator 3 end fraction

fraction numerator straight x plus 2 over denominator negative 1 end fraction space equals space fraction numerator straight y plus 3 over denominator 7 end fraction space equals space fraction numerator straight z minus 3 over denominator begin display style 3 over 2 end style end fraction space space space space space space space or space space space space space fraction numerator straight x plus 2 over denominator negative 2 end fraction space equals space fraction numerator straight y plus 3 over denominator 14 end fraction space equals fraction numerator straight z minus 3 over denominator 3 end fraction

Its direction ratios are –2, 14, 3
∴ cartesian equation of line through (1, 2, 3) and having direction ratios –2, 14, 3 are

fraction numerator straight x minus 1 over denominator negative 2 end fraction space equals space fraction numerator straight y minus 2 over denominator 14 end fraction space equals fraction numerator straight z minus 3 over denominator 3 end fraction

Now,

straight a with rightwards arrow on top space equals space straight i with overparenthesis on top space plus space 2 straight j with overparenthesis on top plus 3 stack straight k comma with overparenthesis on top space space space straight m with rightwards arrow on top space equals space minus 2 space straight i with overparenthesis on top space plus space 14 space straight j with hat on top space plus space 3 space straight k with hat on top

∴ vector equation of line is

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

straight r with rightwards arrow on top space equals space left parenthesis straight i with hat on top plus 2 space straight j with hat on top space plus space 3 space straight k with hat on top right parenthesis plus straight lambda left parenthesis negative 2 space straight i with hat on top plus 14 space straight j with hat on top plus 3 space straight k with hat on top right parenthesis.

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Short Answer Type

38.

Find the equation of line (vector and cartesian both) which is parallel to the vector $2 straight i with hat on top space minus straight j with hat on top plus 3 space straight k with hat on top$ and which passes through the point (5, -2, 4).

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39. Find the vector equation of the line passing through the point (2, –1, –1) which is parallel to the lines 6 x – 2 = 3 y + 1 = 2 z – 2.

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40.

The equation of a line is given by $fraction numerator 4 minus straight x over denominator 2 end fraction space equals space fraction numerator straight y plus 3 over denominator 3 end fraction space equals fraction numerator straight z plus 2 over denominator 6 end fraction$ . Write the direction cosines of a line parallel to the above line.