Three Dimensional Geometry

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Question

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

Solution

Since the line passes through the point (1, 2, 3) and is parallel to the vector

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

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