Find the vector equation of the line through the origin which is perpendicular to the plane $straight r with rightwards arrow on top. space open parentheses straight i with hat on top space minus space 2 space straight j with hat on top space plus space straight k with hat on top close parentheses space equals space 3.$

Solution

The equation of plane is

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

∴ direction-ratios of the normal are 1,–2, 1
Also normal passes through origin (0, 0, 0)
∴ vector equation of line is

straight r with rightwards arrow on top space space equals space left parenthesis 0 space straight i with hat on top space plus space 0 space straight j with hat on top space plus space 0 space straight k with hat on top right parenthesis space plus space straight lambda space left parenthesis straight i with hat on top space minus space 2 space straight j with hat on top space plus space straight k with hat on top right parenthesis

straight r with rightwards arrow on top space equals space straight lambda open parentheses straight i with hat on top space minus space 2 space straight j with hat on top space plus space straight k with hat on top close parentheses space space where space straight lambda space is space straight a space space parameter.

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