Find the equation of the plane which contains the line of intersection of the planes
$\vec{r} . (\hat{i} + 2 \hat{j} + 3 k) - 4 = 0, \vec{r} . (2 \hat{i} + \hat{j} - k) + 5 = 0$ and which is perpendicular to
the plane $\vec{r} . (5 \hat{i} + 3 \hat{j} - 6 k) + 8 = 0$ .

Solution

The equations of the given planes are

$\vec{r} . (\hat{i} + 2 \hat{j} + 3 k) - 4 = 0 . . . . . . . . . . . . . . (i) \vec{r} . (2 \hat{i} + \hat{j} - k) + 5 = 0 . . . . . . . . . . . . . . . (ii)$

The equation of the plane passing through the line of intersection of the given

planes is

$[\vec{r} . (\hat{i} + 2 \hat{j} + 3 k) - 4] + λ [\vec{r} . (2 \hat{i} + \hat{j} - k) + 5] = 0 \vec{r} . [(1 + 2 λ) \hat{i} + (2 + λ) \hat{j} + (3 - λ) k] + (- 4 + 5 λ) = 0 . . . . . . . . . (iii) The plane in equation (iii) is perpendicular to the plane, \vec{r} . (5 \hat{i} + 3 \hat{j} - 6 k) + 8 = 0 ∴ 5 (1 + 2 λ) + 3 (2 + λ) - 6 (3 - λ) = 0 \Rightarrow 5 + 10 λ + 6 + 3 λ - 18 + 6 λ = 0 \Rightarrow 19 λ - 7 = 0 \Rightarrow λ = \frac{7}{19} Substituting λ = \frac{7}{19} in equation (iii),$

$\vec{r} . [\frac{33}{19} \hat{i} + \frac{45}{19} \hat{j} + \frac{50}{19} k] - \frac{41}{19} = 0 \Rightarrow \vec{r} . [33 \hat{i} + 45 \hat{j} + 50 k] - 41 = 0$

This is the vector equation of the required plane.

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Three Dimensional Geometry

Find the equation of the plane which contains the line of intersection of the planes
$\vec{r} . (\hat{i} + 2 \hat{j} + 3 k) - 4 = 0, \vec{r} . (2 \hat{i} + \hat{j} - k) + 5 = 0$ and which is perpendicular to
the plane $\vec{r} . (5 \hat{i} + 3 \hat{j} - 6 k) + 8 = 0$ .

Some More Questions From Three Dimensional Geometry Chapter

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

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For Daily Free Study Material Join wiredfaculty

Three Dimensional Geometry

Find the equation of the plane which contains the line of intersection of the planes r→. i^ + 2 j^ + 3 k - 4 = 0, r→. 2 i^ + j^ - k + 5 = 0 and which is perpendicular tothe plane r→. 5 i^ + 3 j^ - 6 k + 8 = 0.

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

Find the equation of the plane which contains the line of intersection of the planes
$\vec{r} . (\hat{i} + 2 \hat{j} + 3 k) - 4 = 0, \vec{r} . (2 \hat{i} + \hat{j} - k) + 5 = 0$ and which is perpendicular to
the plane $\vec{r} . (5 \hat{i} + 3 \hat{j} - 6 k) + 8 = 0$ .