The Planes

### If the plane 2x ā 3y ā 6z = 13 makes an angle with the x-axis, then find the value of Ī».

Answer: Direction ratios of the x-axis are 1,0,0 and direction ratios of normal to the plane are 2,ā3,ā6. The angle between the line and the plane,...

### 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,ā11 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 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/ā29 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 =...