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Three Dimensional Geometry

Question
CBSEENMA12033373

Find the equation of the plane through the point (3, 4,–1) which is parallel to the plane straight r with rightwards arrow on top. space open parentheses 2 straight i with hat on top space minus space 3 space straight j with hat on top space plus space 5 space straight k with hat on top close parentheses space plus space 7 space equals space 0.

Solution
The equation of plane is
straight r with rightwards arrow on top. space open parentheses 2 space straight i with hat on top space minus space 3 space straight j with hat on top space plus space 5 space straight k with hat on top close parentheses space plus space 7 space equals space 0
Any plane parallel to it is
        straight r with rightwards arrow on top. space open parentheses 2 space straight i with hat on top space minus space 3 space straight j with hat on top space plus space 5 space straight k with hat on top close parentheses space equals space straight d                           ...(1)
∴   point (3, 4, – 1) lies on it
therefore space space space straight r with rightwards arrow on top space equals space 3 space straight i with hat on top space plus space 4 space straight j with hat on top space minus space straight k with hat on top  satisfies (1)
therefore space space space space open parentheses 3 space straight i with hat on top space plus space 4 space straight j with hat on top space minus space straight k with hat on top close parentheses space. space open parentheses 2 space straight i with hat on top space minus space space 3 space straight j with hat on top space plus space 5 space straight k with hat on top close parentheses space equals space straight d
rightwards double arrow space space space space left parenthesis 3 right parenthesis thin space left parenthesis 2 right parenthesis space plus space left parenthesis 4 right parenthesis thin space left parenthesis negative 3 right parenthesis space plus space left parenthesis negative 1 right parenthesis thin space left parenthesis 5 right parenthesis space equals space straight d
rightwards double arrow space space space space 6 minus 12 minus 5 space equals space straight d space space space space rightwards double arrow space straight d space equals space minus 11
Putting this value of d in (1), we get
                       straight r with rightwards arrow on top space. space open parentheses 2 space straight i with hat on top space minus space 3 space straight j with hat on top space plus space 5 space straight k with hat on top close parentheses space equals space minus 11
or                    straight r with rightwards arrow on top. open parentheses 2 space straight i with hat on top space minus space 3 space straight j with hat on top space plus space 5 space straight k with hat on top close parentheses plus 11 space equals space 0 space
which is required equation of plane. 

Some More Questions From Three Dimensional Geometry Chapter

Find the direction cosines of x, y and z-axis.