Find the equation of the plane passing through the intersection of the planes r → . ( i ^ + 3 j ^ - k ^ ) = 9 and r → . ( 2 i ^ - j ^ + k ^ ) = 3 and passing through the origin.

Given equation of the plane are r → . ( i ^ + 3 j ^ - k ^ ) = 9 . . . . . . . . . . . . . . . . i and r → . ( 2 i ^ - j ^ + k ^ ) = 3 . . . . . . . . . . . . . i i In Cartesian form equation of the planes are given by x i + y j + z k . ( i ^ + 3 j ^ - k ^ ) = 9 ⇒ x + 3 y - z = 9 . . . . . . . . . . . . . . . i i i and x i + y j + z k . 2 i - j + k = 3 ⇒ 2 x - y + z = 3 . . . . . . . . . . . . . . . . i v Thus, equation of plane passing through the intersection of planes i i i and i v is given by x + 3 y - z - 9 + k 2 x - y + z - 3 = 0 . . . . . . . . v Since v passes through origin 0 , 0 , 0 ∴ 0 + 0 - 0 - 9 + k 0 - 0 + 0 - 3 = 0 ⇒ - 9 - 3 k = 0 ⇒ k = - 3 Putting the value of k in v we get x + 3 y - z - 9 - 3 2 x - y + z - 3 = 0 ⇒ - 5 x + 6 y - 4 z = 0 ⇒ 5 x - 6 y + 4 z = 0 Which is the required equation of the plane.

Find the vector equation of the plane passing through the intersection of the planes r → • ( 2 i ^ - 7 j ^ + 4 k ^ ) = 3 and r → • ( 3 i ^ - 5 j ^ + 4 k ^ ) + 11 = 0 and passing through the point ( - 2 , 1 , 3 ) .

The vector equation of given planes are r → . ( 2 i ^ - 7 j ^ + 4 k ^ ) = 3 , a n d r → . ( 3 i ^ - 5 j ^ + 4 k ^ ) + 11 = 0 In Cartesian form it will be 2 x - 7 y _ 4 z = 3 . . . . . . . . . . . . . . i 3 x - 5 y + 4 z + 11 = 0 . . . . . . . . . . . i i Thus, the equation of plane passing through the intersection of plane 1 and plane 2 is given by 2 x - 7 y + 4 z + λ 3 x - 5 y + 4 z + 11 = 0 . . . . . . . . i i i Since, equation i i i passing through the point - 2 , 1 , 3 ∴ - 4 - 7 + 12 - 3 + λ - 6 - 5 + 12 + 11 = 0 ⇒ - 2 + λ 12 = 0 ⇒ λ = 1 6 putting the value in i i i we get 2 x - 7 y + 4 z + 1 6 3 x - 5 y + 4 z + 11 = 0 ⇒ 15 x - 47 y + 28 z - 7 = 0 . . . . . . . . i v If r → be the position vector of point P x , y , z then r → = x i + y j + z k Then i v can be written as x i + y j + z k . 15 x - 47 y + 28 z - 7 = 0 ⇒ r → . 15 x - 47 y + 28 z - 7 = 0 Which is the required equation of plane.

Find the equation of the plane passing through the intersection of the planes r → . ( 2 i ^ + j ^ + 3 k ^ ) = 7 and r → • ( 2 i ^ + 5 j ^ + 3 k ^ ) = 9 and the point ( 2 , 1 , 3 ) .

The equation of plane passing through the line of intersection of given planes r → . ( 2 i ^ + j ^ + 3 k ^ ) = 7 and r → . ( 2 i ^ + 5 j ^ + 3 k ^ ) = 9 is given by r → . ( 2 i ^ + j ^ + 3 k ^ ) - 7 + λ r → . ( 2 i ^ + 5 j ^ + 3 k ^ ) - 9 = 0 ⇒ r → . 2 + 2 λ i + 1 + 5 λ j + 3 + 3 λ k - 7 - 9 λ = 0 . . . . . . . i Since the plane passes through the point 2 , 1 , 3 ∴ 2 i + j + 3 k . 2 + 2 λ i + 1 + 5 λ j + 3 + 3 λ k - 7 - 9 λ = 0 ⇒ 2 2 + 2 λ + 1 + 5 λ + 3 3 + 3 λ - 7 - 9 λ = 0 ⇒ 9 λ = - 7 ⇒ λ = - 7 9 Putting the value in i we get r → . 4 9 i - 26 9 j + 6 9 k - 7 + 7 = 0 ⇒ r → . 2 i - 13 j = 3 k = 0 Which is the required equation of plane.

E M B I B E

Understanding ISC Mathematics Class 12 Volume 2>Chapter 2 - Three Dimensional Geometry>EXERCISE

M L Aggarwal Solutions for Chapter: Three Dimensional Geometry, Exercise 6: EXERCISE

Author:M L Aggarwal

M L Aggarwal Mathematics Solutions for Exercise - M L Aggarwal Solutions for Chapter: Three Dimensional Geometry, Exercise 6: EXERCISE

Attempt the free practice questions on Chapter 2: Three Dimensional Geometry, Exercise 6: EXERCISE with hints and solutions to strengthen your understanding. Understanding ISC Mathematics Class 12 Volume 2 solutions are prepared by Experienced Embibe Experts.

Questions from M L Aggarwal Solutions for Chapter: Three Dimensional Geometry, Exercise 6: EXERCISE with Hints & Solutions

EASY

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 2>Chapter 2 - Three Dimensional Geometry>EXERCISE>Q 1

Write the intercept cut off by the plane $2 x + y - z = 5$ on $x$ -axis.

EASY

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 2>Chapter 2 - Three Dimensional Geometry>EXERCISE>Q 1

Write the intercept cut off by the plane $3 x - 4 y + z = 6$ on y-axis.

EASY

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 2>Chapter 2 - Three Dimensional Geometry>EXERCISE>Q 1

Write the intercept cut off by the plane $5 x - 2 y + 3 z + 7 = 0$ on $z$ -axis.

MEDIUM

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 2>Chapter 2 - Three Dimensional Geometry>EXERCISE>Q 24

Find the equation of the plane through the intersection of the planes $3 x - y + 2 z - 4 = 0, x + y + z - 2 = 0$ and the point $(2, 2, 1) .$

MEDIUM

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 2>Chapter 2 - Three Dimensional Geometry>EXERCISE>Q 25

Find the equation of the plane passing through the intersection of the planes $\vec{r} . (\hat{i} + 3 \hat{j} - \hat{k}) = 9$ and $\vec{r} . (2 \hat{i} - \hat{j} + \hat{k}) = 3$ and passing through the origin.

MEDIUM

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 2>Chapter 2 - Three Dimensional Geometry>EXERCISE>Q 25

Find the vector equation of the plane passing through the intersection of the planes $\vec{r} • (2 \hat{i} - 7 \hat{j} + 4 \hat{k}) = 3$ and $\vec{r} • (3 \hat{i} - 5 \hat{j} + 4 \hat{k}) + 11 = 0$ and passing through the point $(- 2, 1, 3) .$

MEDIUM

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 2>Chapter 2 - Three Dimensional Geometry>EXERCISE>Q 25

Find the equation of the plane passing through the intersection of the planes $\vec{r} . (2 \hat{i} + \hat{j} + 3 \hat{k}) = 7$ and $\vec{r} • (2 \hat{i} + 5 \hat{j} + 3 \hat{k}) = 9$ and the point $(2, 1, 3)$ .

MEDIUM

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 2>Chapter 2 - Three Dimensional Geometry>EXERCISE>Q 26

Find the equation of the plane passing through the line of intersection of the planes $x + 2 y + z = 3, 2 x - y - z = 5$ and the point $(2, 1, 3) .$ Also find direction ratios of a normal to this plane.

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