Question

Find the vector equation of the plane passing through the intersection of the planes:
$straight r with rightwards arrow on top. space open parentheses straight i with hat on top space plus space straight j with hat on top space plus space straight k with hat on top close parentheses space equals space 6 comma space space space space straight r with rightwards arrow on top. space space open parentheses 2 space straight i with hat on top space plus space space 3 space straight j with hat on top space plus space 4 space straight k with hat on top close parentheses space equals space minus 5$ and the point (1, 1, 1).

Solution

The equations of given planes are
   $straight r with rightwards arrow on top space. space open parentheses straight i with hat on top space plus space straight j with hat on top space plus space straight k with hat on top close parentheses minus 6 space equals space 0$ ...(1)
and    $straight r with rightwards arrow on top. space open parentheses 2 straight i with hat on top space plus space 3 space straight j with hat on top space plus space 4 space straight k with hat on top close parentheses space plus space 5 space equals 0$ ...(2)
Any plane through the intersection of planes (1) and (2) is
$straight r with rightwards arrow on top. space space open parentheses straight i with hat on top space plus space straight j with hat on top space plus space straight k with hat on top close parentheses space minus space 6 space plus straight lambda space open curly brackets stack straight r. space with rightwards arrow on top open parentheses 2 space straight i with hat on top space plus space 3 space straight j with hat on top space plus space 4 space straight k with hat on top close parentheses space plus space 5 close curly brackets space equals space 0$
or $straight r with rightwards arrow on top. space open curly brackets open parentheses 1 plus 2 space straight lambda close parentheses straight i with hat on top space plus space left parenthesis 1 plus 3 space straight lambda right parenthesis space straight j with hat on top space plus space left parenthesis 1 plus 4 space straight lambda right parenthesis space straight k with hat on top space close curly brackets space plus left parenthesis 5 space straight lambda space minus space 6 right parenthesis space equals space 0$ ...(3)
It will pass through the point (1, 1, 1)
If    $straight r space equals straight i with hat on top space plus space straight j with hat on top space plus space straight k with hat on top space space satisfies space left parenthesis 3 right parenthesis$
$straight i. straight e. space space space if space left parenthesis straight i with hat on top plus straight j with hat on top plus straight k with hat on top right parenthesis. space space open curly brackets left parenthesis 1 plus 2 straight lambda right parenthesis space straight i with hat on top space plus space left parenthesis 1 plus 3 space straight lambda right parenthesis space straight j with hat on top space plus space left parenthesis 1 plus 4 space straight lambda right parenthesis space straight k with hat on top close curly brackets space plus space left parenthesis 5 straight lambda minus 6 right parenthesis space equals space 0 straight i. straight e. space if space left parenthesis 1 right parenthesis thin space left parenthesis 1 plus 2 straight lambda right parenthesis plus space left parenthesis 1 right parenthesis thin space left parenthesis 1 plus 3 space straight lambda right parenthesis space plus space left parenthesis 1 right parenthesis thin space left parenthesis 1 plus 4 space straight lambda right parenthesis space plus space left parenthesis 5 space straight lambda space minus space 6 right parenthesis space equals space 0 straight i. straight e. space if space 1 plus 2 straight lambda plus 1 plus 3 straight lambda plus 1 plus 4 straight lambda plus 5 straight lambda minus 6 space equals space 0 straight i. straight e. space if space 14 straight lambda space equals space 3 straight i. straight e. space if space space straight lambda space equals 3 over 14$

Putting $straight lambda space equals 3 over 14 space in space left parenthesis 3 right parenthesis comma space we space get$
$straight r with rightwards arrow on top. space space open curly brackets open parentheses 1 plus 6 over 14 close parentheses straight i with hat on top space plus space open parentheses 1 plus 9 over 14 close parentheses straight j with hat on top space plus space open parentheses 1 plus 12 over 14 close parentheses space straight k with hat on top close curly brackets space plus space open parentheses 15 over 14 minus 6 close parentheses space equals 0$
or $straight r with rightwards arrow on top. space left parenthesis 20 space straight i with hat on top space plus space 23 space straight j with hat on top space plus space 26 space space straight k with hat on top right parenthesis minus 69 space equals space 0$
or $straight r with rightwards arrow on top. space left parenthesis 20 space straight i with hat on top space plus space 23 space straight j with hat on top space plus space 26 space straight k with hat on top right parenthesis space equals space 69$
which is the required vector equation of the plane.

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Three Dimensional Geometry

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