Three Dimensional Geometry

Question

Find the vector and cartesian equations of the plane which passes through the point (5, 2, – 4) and perpendicular to the line with direction ratios 2, 3, –1.

Solution

Here, $stack straight r subscript 1 with rightwards arrow on top space equals space 5 space straight i with hat on top space plus space 2 space straight j with hat on top space minus space 4 space straight k with hat on top comma space straight n with rightwards arrow on top space equals space 2 space straight i with hat on top space space plus space 3 space straight j with hat on top space minus space straight k with hat on top$
The vector equation of plane is
$left parenthesis straight r with rightwards arrow on top space minus space stack straight r subscript 1 with rightwards arrow on top right parenthesis. space straight n with rightwards arrow on top space equals space 0$
or $open square brackets straight r with rightwards arrow on top space minus space left parenthesis 5 space straight i with hat on top space plus space 2 space stack straight j space with hat on top space minus space 4 space straight k with hat on top right parenthesis close square brackets space. space open square brackets left parenthesis 2 stack straight i space with hat on top space plus space 3 space straight j with hat on top space minus space straight k with hat on top right parenthesis close square brackets space equals space 0$
Take $straight r with rightwards arrow on top space equals space straight x space straight i with hat on top space plus space straight y space straight j with hat on top space plus space straight z space straight k with hat on top$
$therefore space space space space space space open square brackets open parentheses straight x space straight i with hat on top space plus space straight y space straight j with hat on top space plus space straight z space straight k with hat on top close parentheses space minus space left parenthesis 5 space stack straight i space with hat on top space plus space 2 space straight j with hat on top space minus space 4 space straight k with hat on top right parenthesis space close square brackets. space space left parenthesis 2 space straight i with hat on top space plus space 3 space straight j with hat on top space minus space straight k with hat on top right parenthesis space equals space 0$
or $open square brackets left parenthesis straight x minus 5 right parenthesis space straight i with hat on top space space left parenthesis straight y minus 2 right parenthesis space straight j with hat on top space plus space left parenthesis straight z plus 4 right parenthesis space straight k with hat on top close square brackets space. space left parenthesis 2 space straight i with hat on top space plus space 3 space straight j with hat on top space minus space straight k with hat on top right parenthesis space equals space 0$
$space or space space 2 space left parenthesis straight x minus 5 right parenthesis space plus space 3 space left parenthesis straight y minus 2 right parenthesis space minus space 1 space left parenthesis straight z plus 4 right parenthesis space equals space 0 space or space space 2 straight x minus 10 space plus space 3 straight y space minus space 6 space minus space straight z space minus 4 space space equals space 0 space or space space 2 straight x plus 3 straight y minus straight z space equals space 20$
which is the cartesian equation of the plane.

For Daily Free Study Material Join wiredfaculty

Three Dimensional Geometry

Find the vector and cartesian equations of the plane which passes through the point (5, 2, – 4) and perpendicular to the line with direction ratios 2, 3, –1.

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

Loaction