Find the equation of the line which passes through the point (3, 4, 5) and is parallel to the vector $2 straight i with hat on top plus 2 straight j with hat on top minus 3 straight k with hat on top.$

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Solution

Since the line passes through the point (3, 4, 5) and is parallel to the vector

2 straight i with overparenthesis on top plus 2 space straight j with overparenthesis on top space minus space 3 space straight k with overparenthesis on top.

therefore

straight a with rightwards arrow on top space equals space 3 straight i with hat on top plus 4 straight j with hat on top plus 5 straight k with hat on top comma space space straight m with rightwards arrow on top space equals space 2 straight i with hat on top space plus space 2 straight j with hat on top space minus space 3 straight k with hat on top

Let

straight r with rightwards arrow on top

be the position vector of any point on the line.
∴ the vector equation of line is

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

straight r space equals space left parenthesis 3 straight i with hat on top space plus space 4 straight j with hat on top space plus space 5 straight k with hat on top right parenthesis space plus space straight lambda space left parenthesis 2 straight i with hat on top space plus space 2 straight j with hat on top space minus space 3 straight k with hat on top right parenthesis

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