Question

Find the equation of the plane passing through the points (3, 2, 1) and (0, 1, 7) and parallel to the line $straight r with rightwards arrow on top space equals space 2 space straight i with hat on top space minus space straight j with hat on top space plus space straight k with hat on top space plus space straight lambda left parenthesis straight i with hat on top space minus space straight j with hat on top space minus space straight k with hat on top right parenthesis.$

Solution

The given points are (3, 2, 1) and (0, 1, 7)

therefore space space space space straight a 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 plus space straight k with hat on top comma space space straight b with rightwards arrow on top space equals space straight j with hat on top space plus space 7 space straight k with hat on top

The equation of line is

straight r 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 straight k with hat on top space space plus space straight lambda space left parenthesis straight i with hat on top space minus space straight j with hat on top space minus space straight k with hat on top right parenthesis

therefore space space space space straight c with rightwards arrow on top space equals space straight i with hat on top space minus space straight j with hat on top space minus space straight k with hat on top

therefore space space space straight b with rightwards arrow on top space cross times space straight c 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 0 1 7 row 1 cell negative 1 end cell cell negative 1 end cell end table close vertical bar space equals space straight i with hat on top space open vertical bar table row 1 7 row cell negative 1 end cell cell negative 1 end cell end table close vertical bar space minus space straight j with hat on top space open vertical bar table row 0 7 row 1 cell negative 1 end cell end table close vertical bar space plus space straight k with hat on top open vertical bar table row 0 1 row 1 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 space space space space equals space left parenthesis negative 1 plus 7 space right parenthesis straight i with hat on top space minus space left parenthesis 0 minus 7 right parenthesis space straight j with hat on top space plus space left parenthesis 0 minus 1 right parenthesis space straight k with hat on top space equals space 6 space straight i with hat on top space plus space 7 space straight j with hat on top space minus space straight k with hat on top

straight c with rightwards arrow on top cross times straight a 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 cell negative 1 end cell row 3 2 1 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 cell negative 1 end cell row 2 1 end table close vertical bar space minus space straight j with hat on top space open vertical bar table row 1 cell negative 1 end cell row 3 1 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 3 2 end table close vertical bar

equals left parenthesis negative 1 plus 2 right parenthesis space straight i with hat on top space minus space left parenthesis 1 plus 3 right parenthesis space straight j with hat on top space plus space left parenthesis 2 plus 3 right parenthesis space straight k with hat on top space equals space straight i with hat on top space minus space 4 space straight j with hat on top space plus space 5 space straight k with hat on top

therefore space space open square brackets straight a with rightwards arrow on top comma space straight b with rightwards arrow on top comma space straight c with rightwards arrow on top close square brackets space equals space straight a with rightwards arrow on top. space straight b with rightwards arrow on top space cross times space straight c with rightwards arrow on top space equals space open parentheses 3 space straight i with hat on top space plus space 2 space straight j with hat on top space plus space straight k with hat on top close parentheses space. space open parentheses 6 space straight i with hat on top space plus space 7 space straight j with hat on top space minus space straight k with hat on top close parentheses

equals space left parenthesis 3 right parenthesis thin space left parenthesis 6 right parenthesis space plus space left parenthesis 2 right parenthesis thin space left parenthesis 7 right parenthesis space plus space left parenthesis 1 right parenthesis thin space left parenthesis negative 1 right parenthesis space equals space 18 plus 14 minus 1 space equals space 31

Required equation of plane is

open square brackets straight r with rightwards arrow on top comma space straight b with rightwards arrow on top comma space straight c with rightwards arrow on top close square brackets space plus space open square brackets straight r with rightwards arrow on top comma space straight c with rightwards arrow on top comma space straight a with rightwards arrow on top close square brackets space equals space open square brackets straight a with rightwards arrow on top comma space straight b with rightwards arrow on top comma space straight c with rightwards arrow on top close square brackets

or space space straight r with rightwards arrow on top. space open parentheses straight b with rightwards arrow on top space cross times space straight c with rightwards arrow on top close parentheses space plus space straight r with rightwards arrow on top. space open parentheses straight c with rightwards arrow on top space cross times space straight a with rightwards arrow on top close parentheses space equals space open square brackets straight a with rightwards arrow on top comma space straight b with rightwards arrow on top comma space straight c with rightwards arrow on top close square brackets or space space straight r with rightwards arrow on top. space open curly brackets open parentheses straight b with rightwards arrow on top cross times straight c with rightwards arrow on top close parentheses space plus space open parentheses straight c with rightwards arrow on top cross times straight a with rightwards arrow on top close parentheses close curly brackets space equals space open square brackets straight a with rightwards arrow on top comma space straight b with rightwards arrow on top comma space straight c with rightwards arrow on top close square brackets or space space space straight r with rightwards arrow on top. space open parentheses 6 space straight i with hat on top plus 7 stack space straight j with hat on top space minus space straight k with hat on top space plus space straight i with hat on top space minus space 4 space straight j with hat on top space plus space 5 space straight k with hat on top close parentheses space equals space 31 or space space straight r with rightwards arrow on top. space open parentheses 7 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 equals space 31.

For Daily Free Study Material Join wiredfaculty

Three Dimensional Geometry

Some More Questions From Three Dimensional Geometry Chapter

Find the direction cosines of x, y and z-axis.

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

Phone Number

Email Address

Location