Find equation of a line, which is parallel to $2 straight i with hat on top space plus space straight j with hat on top space minus space 3 space straight k with hat on top$ and passes through (1, ,1, 1).

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Solution

We know that the equation of a straight line passing through a fixed point with position vector

straight a with rightwards arrow on top space and space parallel space to space the space vector space straight m with rightwards arrow on top space is space

straight r with rightwards arrow on top space equals space straight a with rightwards arrow on top space plus space straight lambda space straight m with rightwards arrow on top

...(1) where X is a parameter
Here,

straight a with rightwards arrow on top space equals space straight i with hat on top space plus space straight j with hat on top space plus space straight k with hat on top space space space and space space straight m with rightwards arrow on top space equals space 2 straight i with hat on top space plus space straight j with hat on top space minus space 3 space straight k with hat on top

∴ from (1), the vector equation of line is

straight r with rightwards arrow on top space equals space left parenthesis straight i with hat on top plus straight j with hat on top space plus space stack straight k right parenthesis with hat on top space plus space straight lambda space left parenthesis 2 space straight i with hat on top space plus space straight j with hat on top space minus space 3 space straight k with hat on top right parenthesis

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