Answer: Direction ratios of the x-axis are 1,0,0 and direction ratios of normal to the plane are 2,ā3,ā6. The angle between the line and the plane,...

### Find the vector equation of a line passing through the origin and perpendicular to the plane

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### Find the angle between the line and the plane 10x + 2y ā 11z = 3.

Answer: Direction ratios of the given line are 2,3,6 Direction ratios of the normal to the given plane are 10,2,ā11 The angle between the line and the plane: \begin{array}{l} \sin...

### Find the angle between the line and the plane

Answer: The equation of line is: $\vec{r}=(i+2 \hat{j}-\hat{k})+\lambda(\hat{i}-\hat{j}+\hat{k})$ Here, $\vec{b}=\hat{i}-j+\vec{k}$ The equation of plane is: r→·(2i^-j^+k^)=4 \vec{r}...

### Find the angle between the line and the plane

Answer: The equation of plane is: r→=(2i^-j^+3k^)+λ(3i^-j^+2k^) \vec{r}=(2 \hat{i}-\hat{j}+3 \hat{k})+\lambda(3 \hat{i}-\hat{j}+2 \hat{k}) On comparing with $\vec{r}=\vec{a}+\lambda...

### Find the value of m for which the line is parallel to the plane

Answer: Given equation of line, $\overrightarrow rĀ = (\widehat i + 2\widehat k) + \lambda (2\widehat i Ā - m\widehat j - 3\widehat k)$ Comparing with the line $\overrightarrow rĀ = \overrightarrow...

### Find the angle between the line joining the points A(3,ā4,ā2) and B(12,2,0) and the plane 3xāy+z=1.

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### Show that the line line is parallel to the plane . Also, find the distance between them.

Answer: A line $\begin{array}{l} \overrightarrow rĀ = \overrightarrow aĀ + \lambda \overrightarrow b \\ \end{array}$ is parallel to the plane $\begin{array}{l} \overrightarrow r .\overrightarrow nĀ ...

### Find the angle between the line and the plane

Answer: The equation of line is: r→=(3i^+k^)+λ(j^+k^) \vec{r}=(3 \hat{i}+\hat{k})+\lambda(\hat {j}+\hat{k}) Comparing with $\vec{r}=\vec{a}+\lambda \vec{b}$ $\vec{b}=(j+\dot{k})$. The...

### Find the angle between the line

and the plane

Answer: The given line is $\frac{x-2}{3}=\frac{y+1}{-1}=\frac{2-3}{2}$ ⇒r→=(2i→-j→+k→)+t(3i→-j→-2k→) \Rightarrow...

### Find the vector and Cartesian equations of the plane passing through the origin and parallel to the vectors and .

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### Find the vector and Cartesian equations of the plane passing through the point (3, -1, 2) and parallel to the lines .

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### Find the vector equation of a plane passing through the point (1, 2, 3) and parallel to the lines whose direction ratios are 1, -1, -2 and -1, 0, 2.

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### Prove that the lines are coplanar. Also find the equation of the plane containing these lines.

Answer: - 7x + 14 + 14y ā 56 ā 7z + 42 = 0 - 7x + 14y ā 7z = 0 Multiply by negative sign, x ā 2y + z = 0 The equation of...

### Prove that the lines are coplanar. Also find the equation of the plane containing these lines.

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### Find the vector and Cartesian equations of a plane containing the two lines . Also show that the lines lies in the plane.

Answer: ...

### Find the vector and Cartesian forms of the equations of the plane containing the two lines .

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### Show that the lines are coplanar. Also find the equation of the plane containing these lines.

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### Find the vector equation of a plane passing through the point (1, 2, 3) and parallel to the lines whose direction ratios are 1, -1, -2 and -1, 0, 2.

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### Find the vector and Cartesian equations of the plane passing through the point (3, -1, 2) and parallel to the lines

Answer:

### Find the vector and Cartesian equations of the plane passing through the origin and parallel to the vectors and .

Answer:

### Find the acute angle between the following planes: (i) and (ii) and .

Answer: (i) (ii) ...

### Find the acute angle between the following planes: (i) and (ii) and .

Answer: (i) (ii)

### Find the value of Ī» for which the given planes are perpendicular to each other: (i) and (ii) and .

Answer: (i) For plane perpendicular, cos 900 = 0 (ii) For plane perpendicular, cos 900 = 0 (Ī» Ć 1) + (2 Ć...

### Show that the following planes are at right angles: (i) and (ii) and .

Answer: (i) (ii) = 6 + 24 ā 30 = 30 ā 30 = 0 = RHS Hence, the planes are...

### Find the acute angle between the following planes: (i) 2x ā y + z = 5 and x + y + 2z = 7 (ii) x + 2y + 2z = 3 and 2x ā 3y + 6z = 8

Answer: (i) The standard equation of plane, (ii) The standard equation of plane, ...

### Find the acute angle between the following planes: (i) x + y ā z = 4 and x + 2y + z = 9 (ii) x + y ā 2z = 6 and 2x ā 2y + z = 11

Answer: (i) The standard equation of plane, (ii) The standard equation of plane, ...

### Show that each of the following pairs of planes are at right angles: (i) 3x + 4y ā 5z = 7 and 2x + 6y + 6z + 7 = 0 (ii) x ā 2y + 4z = 10 and 18x + 17y + 4z = 49

Answer: (i) if Īø = 900 then cos 900 = 0 A1A2 + B1B2 + C1C2 = 0 By comparing with the standard equation of a plane, A1 = 3, B1 = 4, C1 = -5 A2 = 2, B2 = 6, C2 = 6 LHS = A1A2 + B1B2 + C1C2 = (3 Ć 2) +...

### Find the equation of the plane passing through the intersection of the planes x ā 2y + z = 1 and 2x + y + z = 8, and parallel to the line with direction ratios 1, 2, 1. Also, find the perpendicular distance of (1, 1, 1) from the plane.

Answer: Cartesian form of equation of plane through the line of intersection of planes: ...

### Find the equation of the plane passing through the line of intersection of the planes 2x ā y = 0 and 3z ā y = 0, and perpendicular to the plane 4x + 5y ā 3z = 0.

Answer: Cartesian form of equation of plane through the line of intersection of planes: The equation...

### Find the equation of the plane through the line of intersection of the planes x ā 3y + z + 6 = 0 and x + 2y + 3z + 5 = 0, and passing through the origin.

Answer: Cartesian form of equation of plane through the line of intersection of planes: - x ā 27y ā 13z = 0 Multiplying by negative sign, x + 27y + 13z = 0 The equation of the plane is x + 27y + 13z...

### Find the equation of the planes passing through the intersection of the planes 2x + 3y ā z + 1 = 0 and x + y ā 2z + 3 = 0, and perpendicular to the plane 3x ā y ā 2z ā 4 = 0.

Answer: Cartesian form of equation of plane through the line of intersection of planes: The equation of the plane is 7x + 13y + 4z = 9.

### Find the equation of the plane through the line of intersection of the planes x + y + z = 6 and 2x + 3y + 4z + 5 = 0, and passing through the point (1, 1, 1).

Answer: Cartesian form of equation of plane through the line of intersection of planes: 14x + 14y + 14z ā 84 + 6x + 9y + 12z + 15 = 0 20x + 23y + 26z...

### 5. Find the equation of the plane passing through the point (1, 4, -2) and parallel to the plane 2x ā y + 3z + 7 = 0.

Answer: Any plane parallel to 2x ā y + 3z + 7 = 0 is shown as 2x ā y + 3z + d = 0. It passes through the point (1, 4, -2) $\begin{array}{l} 2 \times 1 - 4 + 3( - 2) + d = 0\\ 2 - 4 - 6 + d = 0\\ d =...

### Find the vector equation of the plane passing through the point (1, 1, 1) and parallel to the plane .

Answer:

### Find the vector equation of the plane passing through the point (a, b, c) and parallel to the plane .

Answer:

### Find the vector equation of the plane through the points and parallel to the plane .

Answer:

### Show that the planes 2x ā y + 6z = 5 and 5x ā 2.5y + 15z = 12 are parallel.

Answer:

### Find the vector and Cartesian equations of a plane which is at a distance of 7 units from the origin and whose normal vector from the origin is

Answer: Given, $\begin{array}{l} d = 7\\ \overline nĀ = 3\widehat i + 5\widehat j - 6\widehat k \end{array}$ The unit vector normal to the plane: ...

### Find the distance of the point (2, 1, 0) from the plane 2x + y ā 2z + 5 = 0.

Answer: Given plane, 2x + y ā 2z + 5 = 0 The point is (2, 1, 0).

### Find the distance of the point (1, 1, 2) from the plane plane .

Answer: Given plane, $\overrightarrow r .(2\widehat i - 2\widehat j + 4\widehat k) + 5 = 0$ The cartesian form: 2x ā 2y + 4z + 5 = 0 The point is (1, 1, 2).

### Find the distance of the point (3, 4, 5) from the plane .

Answer: Given plane, $\overrightarrow r .(2\widehat i - 5\widehat j + 3\widehat k) = 13$ The cartesian form: 2x ā 5y + 3z ā 13 = 0 The point is (3, 4, 5).

### Find the distance of the point from the plane .

Answer: Given plane, $\overrightarrow r .(\widehat i + \widehat j + \widehat k) + 17 = 0$ The cartesian form: x + y + z +17 = 0 The point is $(\widehat i + 2\widehat j + 5\widehat k)$ => (1, 2,...

### Find the distance of the point from the plane .

Answer: Given plane, $\overrightarrow r .(3\widehat i - 4\widehat j + 12\widehat k) = 9$ The cartesian form: 3x ā 4y + 12z ā 9 =0 The point is $(2\widehat i - \widehat j - 4\widehat k)$ => (2,...

### Find the vector and Cartesian equations of a plane which passes through the point (1, 4, 6) and normal vector to the plane is is

Answer: Given, A = (1, 4, 6)

### Find the vector and Cartesian equations of a plane which is at a distance of 6 units from the origin and which has a normal with direction ratios 2, -1, -2.

Answer: Given, d = 6

### Find the vector and Cartesian equations of a plane which is at a distance of 6/ā29 from the origin and whose normal vector from the origin is is

Answer: Given, = (x Ć 2) + (y Ć (-3)) + (z Ć 4) = 2x - 3y + 4z The Cartesian equation...

### Find the vector equation of a plane which is at a distance of 5 units from the origin and which has as the unit vector normal to it.

Answer: Given, $\begin{array}{l} d = 5\\ \widehat n = \widehat k \end{array}$ The equation of plane at 5 units distance from the origin $\begin{array}{l} \widehat n Ā \end{array}$ and as a unit...

### Reduce the equation of the plane 4x ā 3y + 2z = 12 to the intercept form, and hence find the intercepts made by the plane with the coordinate axes.

Answer: Equation of the plane: 4x ā 3y + 2z = 12 $\begin{array}{l} \frac{4}{{12}}x - \frac{3}{{12}}y + \frac{2}{{12}}z = 1\\ \frac{x}{3} + \frac{y}{{ - 4}} + \frac{z}{6} = 1 \end{array}$ It is the...

### Write the equation of the plane whose intercepts on the coordinate axes are 2, – 4 and 5 respectively.

Answer: Given, Coordinate axes are 2, - 4, 5 The equation of the variable plane: The required equation of the plane is 10x ā 5y + 4z = 20.

### Find the equation of the plane passing through each group of points: A (-2, 6, -6), B (-3, 10, -9) and C (-5, 0, -6)

Answer: Given, A (-2, 6, -6) B (-3, 10, -9) C (-5, 0, -6) ...

### 3. Show that the four points A (0, -1, 0), B (2, 1, -1), C (1, 1, 1) and D (3, 3, 0) are coplanar. Find the equation of the plane containing them.

Answer: Given, A (0, -1, 0) B (2, 1, -1) C (1, 1, 1) D (3, 3, 0) 4x ā 3 (y + 1) + 2z = 0 4x ā 3y + 2z ā 3 = 0 Take x = 0, y = 3 and z =...

### Show that the four points A (3, 2, -5), B (-1, 4, -3), C (-3, 8, -5) and D (-3, 2, 1) are coplanar. Find the equation of the plane containing them.

Answer: Let us take, The equation of the plane passing through A (3, 2, -5) a (x ā 3) + b (y ā 2) + c (z + 5) = 0 It passes through the points B (-1, \4, -3) and C (-3, 8, -5) a (1 ā 3) + b (4 ā 2)...

### Find the equation of the plane passing through each group of points: (i) A (2, 2, -1), B (3, 4, 2) and C (7, 0, 6) (ii) A (0, -1, -1), B (4, 5, 1) and C (3, 9, 4)

Answer: (i) Given, A (2, 2, -1) B (3, 4, 2) C (7, 0, 6) ...