English

Find the Vector Equation of a Plane Which is at a Distance of 7 Units from the Origin and Normal to the Vector.`3hati + 5hatj - 6hatk` - Mathematics

Advertisements
Advertisements

Question

Find the vector equation of a plane which is at a distance of 7 units from the origin and normal to the vector.`3hati + 5hatj - 6hatk`

Advertisements

Solution

shaalaa.com
  Is there an error in this question or solution?
Chapter 11: Three Dimensional Geometry - Exercise 11.3 [Page 493]

APPEARS IN

NCERT Mathematics Part 1 and 2 [English] Class 12
Chapter 11 Three Dimensional Geometry
Exercise 11.3 | Q 2 | Page 493

RELATED QUESTIONS

Find the equations of the planes that passes through three points.

(1, 1, 0), (1, 2, 1), (−2, 2, −1)


Find the vector equation of the line passing through (1, 2, 3) and parallel to the planes `vecr = (hati - hatj + 2hatk)  = 5`and `vecr.(3hati + hatj + hatk) = 6`


Find the vector equation of the line passing through the point (1, 2, − 4) and perpendicular to the two lines: 

`(x -8)/3 = (y+19)/(-16) = (z - 10)/7 and (x - 15)/3 = (y - 29)/8 = (z- 5)/(-5)`


Find the vector equation of a plane passing through a point with position vector \[2 \hat{i} - \hat{j} + \hat{k} \] and perpendicular to the vector  \[4 \hat{i} + 2 \hat{j} - 3 \hat{k} .\] 


Find the Cartesian form of the equation of a plane whose vector equation is 

  \[\vec{r} \cdot \left( - \hat{i} + \hat{j}  + 2 \hat{k} \right) = 9\]

 


Find the vector equations of the coordinate planes.

 

Find the vector equation of each one of following planes. 

x + y − z = 5

 


Find the vector and Cartesian equations of a plane passing through the point (1, −1, 1) and normal to the line joining the points (1, 2, 5) and (−1, 3, 1).

 

Determine the value of λ for which the following planes are perpendicular to each other.

\[\vec{r} \cdot \left( \hat{i}  + 2 \hat{j} + 3 \hat{k} \right) = 7 \text{ and }  \vec{r} \cdot \left( \lambda \hat{i} + 2 \hat{j}  - 7 \hat{k}  \right) = 26\]

 


Determine the value of λ for which the following planes are perpendicular to each ot

 2x − 4y + 3z = 5 and x + 2y + λz = 5


Find the equation of the plane passing through the points (2, 2, 1) and (9, 3, 6) and perpendicular to the plane 2x + 6y + 6z = 1.

 

Find the equation of the plane passing through the points whose coordinates are (−1, 1, 1) and (1, −1, 1) and perpendicular to the plane x + 2y + 2z = 5.

 

Find the equation of the plane that contains the point (1, −1, 2) and is perpendicular to each of the planes 2x + 3y − 2z = 5 and x + 2y − 3z = 8.


Find the vector equation of the line through the origin which is perpendicular to the plane  \[\vec{r} \cdot \left( \hat{i} + 2 \hat{j}  + 3 \hat{k}  \right) = 3 .\]

 

If the lines  \[\frac{x - 1}{- 3} = \frac{y - 2}{- 2k} = \frac{z - 3}{2} \text{ and }\frac{x - 1}{k} = \frac{y - 2}{1} = \frac{z - 3}{5}\] are perpendicular, find the value of and, hence, find the equation of the plane containing these lines.


Find the equation of a plane which passes through the point (3, 2, 0) and contains the line  \[\frac{x - 3}{1} = \frac{y - 6}{5} = \frac{z - 4}{4}\] .

 


Find the reflection of the point (1, 2, −1) in the plane 3x − 5y + 4z = 5.

 

Find the distance of the point (1, −2, 3) from the plane x − y + z = 5 measured along a line parallel to  \[\frac{x}{2} = \frac{y}{3} = \frac{z}{- 6} .\]

 


Find the coordinates of the foot of the perpendicular from the point (2, 3, 7) to the plane 3x − y − z = 7. Also, find the length of the perpendicular.


Find the length and the foot of the perpendicular from the point (1, 1, 2) to the plane \[\vec{r} \cdot \left( \hat{i}  - 2 \hat{j}  + 4 \hat{k}  \right) + 5 = 0 .\]

 

Find the coordinates of the foot of the perpendicular and the perpendicular distance of the  point P (3, 2, 1) from the plane 2x − y + z + 1 = 0. Also, find the image of the point in the plane.


Write the equation of the plane parallel to the YOZ- plane and passing through (−4, 1, 0).

 

Write the equation of the plane passing through points (a, 0, 0), (0, b, 0) and (0, 0, c).

 

Write the ratio in which the plane 4x + 5y − 3z = 8 divides the line segment joining the points (−2, 1, 5) and (3, 3, 2).

 

Write the distance of the plane  \[\vec{r} \cdot \left( 2 \hat{i} - \hat{j} + 2 \hat{k} \right) = 12\] from the origin.

  

Write the position vector of the point where the line \[\vec{r} = \vec{a} + \lambda \vec{b}\] meets the plane  \[\vec{r} . \vec{n} = 0 .\]


Find the length of the perpendicular drawn from the origin to the plane 2x − 3y + 6z + 21 = 0.

 

Find the vector equation of the plane, passing through the point (abc) and parallel to the plane \[\vec{r} . \left( \hat{i}  + \hat{j}  + \hat{k}  \right) = 2\]

 

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


Prove that the lines x = py + q, z = ry + s and x = p′y + q′, z = r′y + s′ are perpendicular if pp′ + rr′ + 1 = 0.


Find the equation of a plane which bisects perpendicularly the line joining the points A(2, 3, 4) and B(4, 5, 8) at right angles.


Find the length and the foot of perpendicular from the point `(1, 3/2, 2)` to the plane 2x – 2y + 4z + 5 = 0.


Find the equation of the plane through the points (2, 1, –1) and (–1, 3, 4), and perpendicular to the plane x – 2y + 4z = 10.


The locus represented by xy + yz = 0 is ______.


The method of splitting a single force into two perpendicular components along x-axis and y-axis is called as ______.


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×