Three Dimensional Geometry

Find the equation of plane containing the coplanar lines - Wired Faculty

Question

Find the equation of plane containing the coplanar lines
$straight r with rightwards arrow on top space equals space straight i with hat on top space plus space straight j with hat on top space plus space straight k with hat on top space plus space straight lambda space open parentheses straight i with hat on top space minus space straight j with hat on top space plus space straight k with hat on top close parentheses space space and space straight r with rightwards arrow on top space equals space 4 space straight i with hat on top space plus space 2 straight k with hat on top space plus space straight mu open parentheses 2 space straight i with hat on top space minus space straight j with hat on top space plus space 3 space straight k with hat on top close parentheses.$

Solution

The equation of given lines are

straight r 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 plus space straight lambda open parentheses straight i with hat on top space minus space straight j with hat on top space plus space straight k with hat on top close parentheses

...(1)
and

straight r with rightwards arrow on top space equals space 4 space straight i with hat on top space plus space 2 space straight k with hat on top space plus space straight mu open parentheses 2 straight i with hat on top space minus space straight j with hat on top space plus space 3 space straight k with hat on top close parentheses

...(2)

therefore space space space space 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 comma space space space straight b with rightwards arrow on top space equals space straight i with hat on top space minus space straight j with hat on top space plus space straight k with hat on top comma space space straight d with rightwards arrow on top space equals space 2 space straight i with hat on top space minus space straight j with hat on top space plus space 3 space straight k with hat on top

therefore space space straight b with rightwards arrow on top space cross times straight d with rightwards arrow on top space equals space open vertical bar table row cell straight i with hat on top end cell cell straight j with hat on top end cell cell straight k with hat on top end cell row 1 cell negative 1 end cell 1 row 2 cell negative 1 end cell 3 end table close vertical bar space equals space straight i with hat on top space open vertical bar table row cell negative 1 end cell 1 row cell negative 1 end cell 3 end table close vertical bar space minus space straight j with hat on top space open vertical bar table row 1 1 row 2 3 end table close vertical bar space plus space straight k with hat on top open vertical bar table row 1 cell negative 1 end cell row 2 cell negative 1 end cell end table close vertical bar space space space space space space space space space space space space space space space space equals space left parenthesis negative 3 space plus space 1 right parenthesis straight i with hat on top space minus space left parenthesis 3 minus 2 right parenthesis space straight j with hat on top space plus space left parenthesis negative 1 plus 2 right parenthesis space straight k with hat on top space equals space minus space 2 straight i with hat on top space minus space straight j with hat on top space plus space straight k with hat on top

Also,

straight a with rightwards arrow on top. space left parenthesis straight b with rightwards arrow on top cross times straight d with rightwards arrow on top right parenthesis space equals space left parenthesis straight i with hat on top space plus space straight j with hat on top space plus space straight k with hat on top right parenthesis. space left parenthesis negative 2 space straight i with hat on top space minus space straight j with hat on top space plus space straight k with hat on top right parenthesis

equals space left parenthesis 1 right parenthesis thin space left parenthesis negative 2 right parenthesis plus left parenthesis 1 right parenthesis thin space left parenthesis negative 1 right parenthesis space plus space left parenthesis 1 right parenthesis thin space left parenthesis 1 right parenthesis space equals space minus 2 minus 1 plus 1 space equals space minus 2

The vector equation of plane containing two lines (1) and (2) is

open square brackets straight r with rightwards arrow on top space minus space straight a with rightwards arrow on top comma space space straight b with rightwards arrow on top comma space space straight d with rightwards arrow on top close square brackets space equals space 0 space space space space space space space space or space space space left parenthesis straight r with rightwards arrow on top space minus space straight a with rightwards arrow on top right parenthesis. space space space straight b with rightwards arrow on top space space cross times space space straight d with rightwards arrow on top space equals space 0

straight r with rightwards arrow on top. space open parentheses straight b with rightwards arrow on top space cross times space straight d with rightwards arrow on top close parentheses space space minus space straight a with rightwards arrow on top. space space open parentheses straight b with rightwards arrow on top cross times straight d with rightwards arrow on top close parentheses space equals space 0

or space space space straight r with rightwards arrow on top. space open parentheses negative 2 space straight i with hat on top space minus space straight j with hat on top space plus space straight k with hat on top close parentheses plus 2 space equals space 0 space space space space or space space space space straight r with rightwards arrow on top. space open parentheses negative 2 space straight i with hat on top space minus space straight j with hat on top space plus space straight k with hat on top close parentheses space equals space minus 2 or space space space straight r with rightwards arrow on top. space open parentheses 2 straight i with hat on top space plus space straight j with hat on top space minus space straight k with hat on top close parentheses space equals space 2

Note. Cartesian equation of plane is

open parentheses straight x space straight i with hat on top space plus space straight y space straight j with hat on top space plus space straight z space straight k with hat on top close parentheses. space space open parentheses 2 space straight i with hat on top space plus space straight j with hat on top space minus space straight k with hat on top close parentheses space equals 2 space space space or space space space 2 straight x plus straight y minus straight z space equals space 2.