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The equation of plane is
2x + 3y – z = 5
Dividing both sides by

square root of left parenthesis 2 right parenthesis squared plus left parenthesis 3 right parenthesis squared plus left parenthesis negative 1 right parenthesis squared end root space equals space square root of 4 plus 9 plus 1 end root space equals space square root of 14 comma space space we space get comma

fraction numerator 2 over denominator square root of 14 end fraction straight x plus fraction numerator 3 over denominator square root of 14 end fraction straight y space minus space fraction numerator 1 over denominator square root of 14 end fraction straight z space equals space fraction numerator 5 over denominator square root of 14 end fraction

It is of form lx + my + nz = p where

straight l equals space fraction numerator 2 over denominator square root of 14 end fraction comma space space straight m space equals space fraction numerator 3 over denominator square root of 14 end fraction comma space space straight n space equals space minus fraction numerator 1 over denominator square root of 14 end fraction comma space space straight p space equals space fraction numerator 5 over denominator square root of 14 end fraction

∴ direction cosines of the normal to the plane are

fraction numerator 2 over denominator square root of 14 end fraction comma space fraction numerator 3 over denominator square root of 14 end fraction comma space minus fraction numerator 1 over denominator square root of 14 end fraction

and distance from origin =

fraction numerator 5 over denominator square root of 14 end fraction.

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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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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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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

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169. Find the vector equation of a plane which is at a distance of 7 units from the origin and which is normal to the vector

3 straight i with hat on top space plus space 5 space straight j with hat on top space minus space 6 space straight k with hat on top.

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170. Find the vector equation of the plane which is at a distance of

fraction numerator 6 over denominator square root of 29 end fraction

from the origin and its normal vector from the origin is

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

Also find its cartesian form.

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