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Three Dimensional Geometry

Short Answer Type

161.

In each of the following cases, determine the direction cosines of the normal to the plane and the distance from the origin.
2x + 3 y – z = 5

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162.

In each of the following cases, determine the direction cosines of the normal to the plane and the distance from the origin.
5y + 8 = 0

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Long Answer Type

163.

In the following cases, find the coordinates of the foot of the perpendicular
drawn from the origin.
2x + 3y + 4z – 12 = 0

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164. In the following cases, find the coordinates of the foot of the perpendicular drawn from the origin.
3 y + 4 z – 6 = 0

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Short Answer Type

165. In the following cases, find the coordinates of the foot of the perpendicular drawn from the origin.
x + y + z = 1

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166. In the following cases, find the coordinates of the foot of the perpendicular drawn from the origin.
5y + 8 = 0

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167. Find the direction cosines of the perpendicular from the origin to the plane

straight r with rightwards arrow on top. space open parentheses 6 space straight i with hat on top space minus space 3 space straight j with hat on top space minus space 2 space straight k with hat on top close parentheses space plus space 1 space equals 0

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168. Find the vector equation of the plane which is at a distance of 5 units from the origin and which is normal to the vector $2 space straight i with hat on top space plus space 6 space straight j with hat on top space minus space 3 space straight k with hat on top$ .

Let $straight n with rightwards arrow on top space equals space 2 space straight i with hat on top space plus space 6 space straight j with hat on top space minus space 3 space straight k with hat on top$
$therefore space space space space space straight n with hat on top space equals space fraction numerator straight n with rightwards arrow on top over denominator open vertical bar straight n with rightwards arrow on top close vertical bar end fraction space equals space fraction numerator 2 straight i with hat on top space plus space 6 straight j with hat on top space minus space 3 space straight k with hat on top over denominator square root of 4 plus 36 plus 9 end root end fraction space equals space 1 over 7 open parentheses 2 straight i with hat on top space plus space 6 straight j with hat on top space minus space 3 straight k with hat on top close parentheses$

$equals space 2 over 7 straight i with hat on top space plus space 6 over 7 straight j with hat on top space minus space 3 over 7 straight k with hat on top$
∴ required equation of plane is

$straight r with rightwards arrow on top. space space open parentheses 2 over 7 straight i with hat on top space plus space 6 over 7 straight j with hat on top space minus space 3 over 7 straight k with hat on top close parentheses space equals 5$

or $straight r with rightwards arrow on top space. space open parentheses 2 space straight i with hat on top space plus space 6 space straight j with hat on top space minus space 3 space straight k with hat on top close parentheses space equals space 35$