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Three Dimensional Geometry

Question
CBSEENMA12033385

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

Solution
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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