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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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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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The equation of plane is

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

...(1)
Now,

open vertical bar 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 vertical bar space space equals space square root of 36 plus 9 plus 4 end root space equals space 7

therefore space space space 6 over 7 straight i with hat on top space minus space 3 over 7 straight j with hat on top space minus space 2 over 7 straight k with hat on top space is space straight a space unit space vector. space

∴ equation (1) of the plane can be written as

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

which is of the form

space straight r with rightwards arrow on top. space straight n with rightwards arrow on top space equals space straight p

∴ perpendicular vector from the origin to the plane is

straight n with rightwards arrow on top space equals space minus 6 over 7 straight i with hat on top space plus space 3 over 7 straight j with hat on top space plus space 2 over 7 straight k with hat on top

therefore

direction cosines of

straight n with rightwards arrow on top space space are space minus 6 over 7 comma space 3 over 7 comma space 2 over 7

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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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