Find the angle between the planes2x – y + z = 6 and x + y + 2z

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 Multiple Choice QuestionsShort Answer Type

141. If O be the origin and the coordinates of P be (1, 2, -3), then find the equation of the plane passing through P and perpendicular to OP.
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144. Find the vector and cartesian equations of the plane which passes through the point (5, 2, – 4) and perpendicular to  the line with direction ratios 2, 3, –1.
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145. Find the cartesian equation of the plane through the point with position vector 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 and perpendicular to the vector 4 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.
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146.

Find the equation of a plane which passes through (2, –3, 1) and is perpendicular to the line through the points (3, 4, –1) and (2,–1, 5).

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147. Find the cartesian equation of the following plane:
straight r with rightwards arrow on top. space open parentheses straight i with hat on top space plus space straight j with hat on top space minus space straight k with hat on top close parentheses space equals space 2
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148. Find the cartesian equation of the following plane:
 straight r with rightwards arrow on top. space open parentheses 2 space 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 close parentheses space equals space 1

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149. Find the cartesian equation of the following plane:
straight r with rightwards arrow on top. space open square brackets left parenthesis straight s minus 2 space straight t right parenthesis space straight i with hat on top space plus space left parenthesis 3 minus straight t right parenthesis space straight j with hat on top space plus space left parenthesis 2 straight s plus straight t right parenthesis space straight k with hat on top close square brackets space equals space 15
 

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150.

Find the angle between the planes
2x – y + z = 6 and x + y + 2z = 7


The equation of two planes are
2 x – y + z = 6
and x + y + 2 z = 7
Let θ be the angle between the planes
therefore space space space space space space cos space straight theta space equals space fraction numerator left parenthesis 2 right parenthesis thin space left parenthesis 1 right parenthesis space plus space left parenthesis negative 1 right parenthesis thin space left parenthesis 1 right parenthesis space plus space left parenthesis 1 right parenthesis thin space left parenthesis 2 right parenthesis over denominator square root of 4 plus 1 plus 1 end root space square root of 1 plus 1 plus 4 end root end fraction
space space space space space space space space space space space space space space space space space space space space equals space fraction numerator 2 minus 1 plus 2 over denominator square root of 6 space square root of 6 end fraction space equals space 3 over 6 space equals space 1 half
therefore space space space space space space space straight theta space space equals space straight pi over 3

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