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

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एनसीईआरटी Mathematics Part 1 and 2 [English] Class 12
अध्याय 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 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 each one of following planes. 

x + y − z = 5

 


A plane passes through the point (1, −2, 5) and is perpendicular to the line joining the origin to the point

\[ \text{ 3 } \hat{i} + \hat{j} - \hat{k} .\] Find the vector and Cartesian forms of the equation of the plane.

 


Find the vector equation of a plane which is at a distance of 5 units from the origin and which is normal to the vector  \[\hat{i}  - \text{2 } \hat{j}  -  \text{2 } \hat{k} .\]

 


find the equation of the plane passing through the point (1, 2, 1) and perpendicular to the line joining the points (1, 4, 2) and (2, 3, 5). Find also the perpendicular distance of the origin from this plane


Find the vector equation of the plane passing through the points P (2, 5, −3), Q (−2, −3, 5) and R (5, 3, −3).


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

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

 


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

 

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

 

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

 

Find the equation of the plane through the points (2, 2, −1) and (3, 4, 2) and parallel to the line whose direction ratios are 7, 0, 6.

 

Find the vector equation of the line passing through (1, 2, 3) and perpendicular to the plane \[\vec{r} \cdot \left( \hat{i}  + 2 \hat{j}  - 5 \hat{k}  \right) + 9 = 0 .\]

 

Find the coordinates of the point where the line through (5, 1, 6) and (3, 4, 1) crosses the  zx - plane .


Find the coordinates of the point where the line through (3, −4, −5) and (2, −3, 1) crosses the plane 2x + y + z = 7.

 

Find the image of the point (0, 0, 0) in the plane 3x + 4y − 6z + 1 = 0.

 

Find the coordinates of the foot of the perpendicular drawn from the point (5, 4, 2) to the line \[\frac{x + 1}{2} = \frac{y - 3}{3} = \frac{z - 1}{- 1} .\]

 Hence, or otherwise, deduce the length of the perpendicular.

 
 

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

 

Write the equation of the plane parallel to XOY- plane and passing through the point (2, −3, 5).

 

Write the intercepts made by the plane 2x − 3y + 4z = 12 on the coordinate axes.

 

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 equation of the plane  \[\vec{r} = \vec{a} + \lambda \vec{b} + \mu \vec{c}\]   in scalar product form.

 

Write the equation of the plane passing through (2, −1, 1) and parallel to the plane 3x + 2y −z = 7.


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

 

The equation of the plane parallel to the lines x − 1 = 2y − 5 = 2z and 3x = 4y − 11 = 3z − 4 and passing through the point (2, 3, 3) is


Find the vector and Cartesian equations of the plane that passes through the point (5, 2, −4) and is perpendicular to the line with direction ratios 2, 3, −1.


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 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 image of the point (1, 6, 3) in the line `x/1 = (y - 1)/2 = (z - 2)/3`.


Two systems of rectangular axis have the same origin. If a plane cuts them at distances a, b, c and a′, b′, c′, respectively, from the origin, prove that `1/"a"^2 + 1/"b"^2 + 1/"c"^2 = 1/"a'"^2 + 1/"b'"^2 + 1/"c'"^2`


Find the foot of perpendicular from the point (2, 3, –8) to the line `(4 - x)/2 = y/6 = (1 - z)/3`. Also, find the perpendicular distance from the given point to the line.


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


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 ______.


A unit vector perpendicular to the plane ABC, where A, B and C are respectively the points (3, –1, 2), (1, –1, –3) and (4, –3, 1), is


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