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Three Dimensional Geometry

Short Answer Type

51. Find a vector equation of the line through the points A (3, 4, – 7) and B (1, – 1, 6).

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52. Find the vector equation for the line passing through the points (– 1, 0, 2) and (3, 4, 6).

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53. Find the vector and cartesian equations of the line that passes through the origin and (5,- 2, 3).

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54. Find the vector and the cartesian equations of the line that passes through the points (3, – 2, – 5), (3, – 2, 6).

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55. Find the vector equation of line joining the points whose vectors are $2 space straight i with hat on top minus straight j with hat on top plus straight k with hat on top space space and space straight i with hat on top space plus space 2 space straight j with hat on top space minus space 3 space straight k with hat on top.$

Here, $straight a with rightwards arrow on top space equals space 2 space straight i with hat on top space minus space straight j with hat on top space plus space straight k with hat on top comma space space space space straight b with rightwards arrow on top space equals space straight i with hat on top space plus space 2 space straight j with hat on top space minus space 3 space straight k with hat on top$
The vector equation of line is
$straight r with rightwards arrow on top space equals space straight a with rightwards arrow on top space plus space straight lambda space left parenthesis straight b with rightwards arrow on top space minus space straight a with rightwards arrow on top right parenthesis$
or $straight r with rightwards arrow on top space equals space left parenthesis 2 straight i with hat on top space minus space straight j with hat on top space plus space straight k with hat on top right parenthesis space plus space straight lambda space left parenthesis negative straight i with hat on top space plus space 3 space straight j with hat on top space minus space 4 space straight k with hat on top right parenthesis$

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56. If the points (–1, 3, 2), (– 4, 2, –2) and (5, 5, λ) are collinear, then find the value of λ.

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Long Answer Type

57. Show that the points whose position vectors are given by

2 space straight i with hat on top space plus space straight j with hat on top space plus space 3 space straight k with hat on top comma space space minus space 4 space straight i with hat on top space plus space 3 space straight j with hat on top space minus space straight k with hat on top comma space space 5 space straight i with hat on top space plus space 5 space straight k with hat on top

are collinear.

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Short Answer Type

58.

Show that the point whose position vectors are given by $negative 2 space straight i with hat on top space plus space 3 space straight j with hat on top space plus space 5 space straight k with hat on top comma space space straight i with hat on top space plus space 2 space straight j with hat on top space plus space 3 space straight k with hat on top space space and space space 7 straight i with hat on top space minus space straight k with hat on top$ are collinear.

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Long Answer Type

59. Find the coordinates of the point where the line through A(3, 4, 1) and B (5, 1, 6) crosses the x y-plane.

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60. Find the coordinates of the point where the line through (5, 1, 6) and (3, 4, 1) crosses the YZ-plane.

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