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Find the Equation of the Plane Passing Through (A, B, C) and Parallel to the Plane Vecr.(Hati + Hatj + Hatk) = 2 - Mathematics

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प्रश्न

Find the equation of the plane passing through (abc) and parallel to the plane `vecr.(hati + hatj + hatk) = 2`

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उत्तर

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पाठ 11: Three Dimensional Geometry - Exercise 11.4 [पृष्ठ ४९८]

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एनसीईआरटी Mathematics Part 1 and 2 [English] Class 12
पाठ 11 Three Dimensional Geometry
Exercise 11.4 | Q 8 | पृष्ठ ४९८

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संबंधित प्रश्‍न

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`


Find the equation of the plane passing through the point (−1, 3, 2) and perpendicular to each of the planes x + 2+ 3z = 5 and 3x + 3z = 0.


If O be the origin and the coordinates of P be (1, 2, −3), then find the equation of the plane passing through P and perpendicular to OP.


Find the vector equation of each one of following planes. 

2x − y + 2z = 8


Show that the normals to the following pairs of planes are perpendicular to each other. 

x − y + z − 2 = 0 and 3x + 2y − z + 4 = 0 


Show that the normals to the following pairs of planes are perpendicular to each other.

\[\vec{r} \cdot \left( 2 \hat{i}  - \hat{j}  + 3 \hat{k}  \right) = 5 \text{ and }  \vec{r} \cdot \left( 2 \hat{i}  - 2 \hat{j}  - 2 \hat{k}  \right) = 5\]

Show that the normal vector to the plane 2x + 2y + 2z = 3 is equally inclined to the coordinate axes.

 

Find the vector equation of a plane which is at a distance of 3 units from the origin and has \[\hat{k}\] as the unit vector normal to it.


Find the vector equation of the plane passing through points A (a, 0, 0), B (0, b, 0) and C(0, 0, c). Reduce it to normal form. If plane ABC is at a distance p from the origin, prove that \[\frac{1}{p^2} = \frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2} .\]

 


Find the vector equation of the plane passing through the points \[3 \hat{i}  + 4 \hat{j}  + 2 \hat{k} , 2 \hat{i} - 2 \hat{j} - \hat{k}  \text{ and }  7 \hat{i}  + 6 \hat{k}  .\]

 

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

 2x − 4y + 3z = 5 and x + 2y + λz = 5


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

 3x − 6y − 2z = 7 and 2x + y − λz = 5

 

Find the equation of a plane passing through the point (−1, −1, 2) and perpendicular to the planes 3x + 2y − 3z = 1 and 5x − 4y + z = 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 (abc) and parallel to the plane  \[\vec{r} \cdot \left( \hat{i}  + \hat{j}  + \hat{k}  \right) = 2 .\]

 

Find the equation of the plane passing through the point (−1, 3, 2) and perpendicular to each of the planes x + 2y + 3z = 5 and 3x + 3y + z = 0.

 

Find the equation of the plane through (2, 3, −4) and (1, −1, 3) and parallel to x-axis.

 

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 coordinates of the foot of the perpendicular from the point (1, 1, 2) to the plane 2x − 2y + 4z + 5 = 0. Also, find the length of the perpendicular.

 

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.


Find the coordinates of the foot of the perpendicular drawn from the origin to the plane 2x − 3y + 4z − 6 = 0.


Find the equation of the plane that contains the point (1, –1, 2) and is perpendicular to both the planes 2x + 3y – 2z = 5 and x + 2y – 3z = 8. Hence, find the distance of point P (–2, 5, 5) from the plane obtained


Find the distance of the point P (–1, –5, –10) from the point of intersection of the line joining the points A (2, –1, 2) and B (5, 3, 4) with the plane x – y + z = 5.


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

 

Write the value of k for which the planes x − 2y + kz = 4 and 2x + 5y − z = 9 are perpendicular.

 

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

 

The vector equation of the plane containing the line \[\vec{r} = \left( - 2 \hat{i} - 3 \hat{j}  + 4 \hat{k}  \right) + \lambda\left( 3 \hat{i}  - 2 \hat{j}  - \hat{k}  \right)\] and the point  \[\hat{i}  + 2 \hat{j}  + 3 \hat{k} \]  is 

 

Find the vector equation of the plane with intercepts 3, –4 and 2 on x, y and z-axis respectively.


Find the value of λ for which the following lines are perpendicular to each other `("x"-5)/(5λ+2) = (2 -"y")/(5) = (1 -"z")/(-1); ("x")/(1) = ("y"+1/2)/(2λ) = ("z" -1)/(3)`

hence, find whether the lines intersect or not


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`


Find the coordinates of the foot of the perpendicular Q  drawn from P(3, 2, 1) to the plane 2x − y + z + 1 = 0. Also, find the distance PQ and the image of the point P treating this plane as a mirror

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


If the foot of perpendicular drawn from the origin to a plane is (5, – 3, – 2), then the equation of plane is `vec"r".(5hat"i" - 3hat"j" - 2hat"k")` = 38.


The point at which the normal to the curve y = `"x" + 1/"x", "x" > 0` is perpendicular to the line 3x – 4y – 7 = 0 is:


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


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