Question

Find the vector and Cartesian form of the equation of the plane passing through the point (1, 2, –4) and parallel to the lines
$straight r with rightwards arrow on top space equals straight i with hat on top space plus space 2 space straight j with hat on top space minus space 4 space straight k with hat on top space plus space straight lambda open parentheses 2 straight i with hat on top space plus space 3 space straight j with hat on top space plus space 6 space straight k with hat on top close parentheses$
and $straight r with rightwards arrow on top space equals space straight i with hat on top space minus space 3 space straight j with hat on top space plus space space 5 space straight k with hat on top space plus space straight mu space open parentheses straight i with hat on top space plus space straight j with hat on top space minus space straight k with hat on top close parentheses$

Solution

Here, $straight a with rightwards arrow on top space equals space straight i with hat on top space plus space 2 space straight j with hat on top space minus space 4 straight k with hat on top comma space space space straight b with rightwards arrow on top space equals space 2 straight i with hat on top space plus space 3 space straight j with hat on top space plus space 6 space straight k with hat on top comma space space straight c with rightwards arrow on top space equals space straight i with hat on top space plus space straight j with hat on top space minus space straight k with hat on top$
The vector equation of plane is
$open parentheses straight r with rightwards arrow on top minus straight a with rightwards arrow on top close parentheses. space open parentheses straight b with rightwards arrow on top space cross times space straight c with rightwards arrow on top close parentheses space equals 0$
$or space space space space open square brackets straight r with rightwards arrow on top space minus space left parenthesis straight i with hat on top space plus space 2 space straight j with hat on top space minus space 4 space straight k with hat on top right parenthesis close square brackets space. space open square brackets left parenthesis 2 space straight i with hat on top space plus space 3 space straight j with hat on top space plus space 6 space straight k with hat on top right parenthesis space cross times space left parenthesis straight i with hat on top space plus space straight j with hat on top space minus space straight k with hat on top right parenthesis close square brackets space equals space 0$
Cartesian equation of plane is
$open vertical bar table row cell straight x minus 1 end cell cell straight y minus 2 end cell cell straight z plus 4 end cell row 2 3 6 row 1 1 cell negative 1 end cell end table close vertical bar space equals space 0$
or $left parenthesis straight x minus 1 right parenthesis space open vertical bar table row 3 6 row 1 cell negative 1 end cell end table close vertical bar space minus space left parenthesis straight y minus 2 right parenthesis space open vertical bar table row 2 6 row 1 cell negative 1 end cell end table close vertical bar space plus space left parenthesis straight z plus 4 right parenthesis space open vertical bar table row 2 3 row 1 1 end table close vertical bar space equals space 0$

or    (–3 – 6) (x – 1) – (–2 – 6) (y – 2) + (2 – 3) (z + 4) = 0
or    –9 (x– 1) + 8 (y – 2) – (z + 4) = 0
or    –9x + 9 + 8y –16-z – 4 = 0
or    –9 x + 8 y – z – 11 = 0 or 9 x – 8 y + z + 11 = 0.

For Daily Free Study Material Join wiredfaculty

Three Dimensional Geometry

Some More Questions From Three Dimensional Geometry Chapter

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

Phone Number

Email Address

Location