मराठी

Find the Vector Equation of the Plane Passing Through the Intersection of the Planes `Vecr.(2hati + 2hatj - 3hatk) = 7, Vecr.(2hati + 5hatj + 3hatk) = 9` and Through the Point (2, 1, 3) - Mathematics

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प्रश्न

Find the vector equation of the plane passing through the intersection of the planes `vecr.(2hati + 2hatj - 3hatk) = 7, vecr.(2hati + 5hatj + 3hatk) = 9` and through the point (2, 1, 3)

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उत्तर

This is the vector equation of the required plane.

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पाठ 11: Three Dimensional Geometry - Exercise 11.3 [पृष्ठ ४९३]

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एनसीईआरटी Mathematics Part 1 and 2 [English] Class 12
पाठ 11 Three Dimensional Geometry
Exercise 11.3 | Q 10 | पृष्ठ ४९३

संबंधित प्रश्‍न

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


Find the equation of the plane passing through the line of intersection of the planes `vecr.(hati + hatj + hatk) = 1` and `vecr.(2hati + 3hatj -hatk) + 4 = 0` and parallel to x-axis.


Find the equation of the plane which contains the line of intersection of the planes `vecrr.(hati + 2hatj + 3hatk) - 4 = 0, vecr.(2hati + htj - hatk) + 5 = 0`,  and which is perpendicular to the plane `vecr.(5hati + 3hatj - 6hatk) + 8 = 0`.


Find the coordinates of the point where the line  \[\frac{x - 2}{3} = \frac{y + 1}{4} = \frac{z - 2}{2}\]   intersects the plane x − y + z − 5 = 0. Also, find the angle between the line and the plane. 

 

Find the distance of the point (−1, −5, −10) from the point of intersection of the line \[\vec{r} = \left( 2 \hat{i}  - \hat{j} + 2 \hat{k}  \right) + \lambda\left( 3 \hat{i}+ 4 \hat{j} + 2 \hat{k}  \right)\] and the plane  \[\vec{r} . \left( \hat{i}  - \hat{j}  + \hat{k} \right) = 5 .\]

 

Find the distance of the point (2, 12, 5) from the point of intersection of the line \[\vec{r} = 2 \hat{i}  - 4 \hat{j}+ 2 \hat{k}  + \lambda\left( 3 \hat{i}  + 4 \hat{j}  + 2 \hat{k} \right)\] and \[\vec{r} . \left( \hat{i}  - 2 \hat{j}  + \hat{k}  \right) = 0\]

  

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  \[x - y + z = 5\] . 

 


Find the distance of the point (1, -5, 9) from the plane

\[x - y + z =\] 5  measured along the line \[x = y = z\]  . 
 

Find the equation of the plane which contains two parallel lines\[\frac{x - 4}{1} = \frac{y - 3}{- 4} = \frac{z - 2}{5}\text{  and }\frac{x - 3}{1} = \frac{y + 2}{- 4} = \frac{z}{5} .\]


Show that the lines  \[\frac{x + 4}{3} = \frac{y + 6}{5} = \frac{z - 1}{- 2}\] and 3x − 2y + z + 5 = 0 = 2x + 3y + 4z − 4 intersect. Find the equation of the plane in which they lie and also their point of intersection.

  

Find the vector equation of the plane passing through three points with position vectors  \[\hat{i}  + \hat{j}  - 2 \hat{k}  , 2 \hat{i}  - \hat{j}  + \hat{k}  \text{ and }  \hat{i}  + 2 \hat{j}  + \hat{k}  .\]  Also, find the coordinates of the point of intersection of this plane and the line  \[\vec{r} = 3 \hat{i}  - \hat{j}  - \hat{k}  + \lambda\left( 2 \hat{i}  - 2 \hat{j} + \hat{k} \right) .\]

 

The plane 2x − (1 + λ) y + 3λz = 0 passes through the intersection of the planes


The equation of the plane through the intersection of the planes x + 2y + 3z = 4 and 2x + y − z = −5 and perpendicular to the plane 5x + 3y + 6z + 8 = 0 is



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A vector parallel to the line of intersection of the planes\[\vec{r} \cdot \left( 3 \hat{i} - \hat{j} + \hat{k}  \right) = 1 \text{ and }  \vec{r} \cdot \left( \hat{i} + 4 \hat{j}  - 2 \hat{k}  \right) = 2\] is 

 

The distance of the point (−1, −5, −10) from the point of intersection of the line \[\vec{r} = 2 \hat{i}- \hat{j} + 2 \hat{k}  + \lambda\left( 3 \hat{i}  + 4 \hat{j}+ 12 \hat{k}  \right)\]   and the plane \[\vec{r} \cdot \left( \hat{i} - \hat{j} + \hat{k}  \right) = 5\] is 

 
 

The equation of the plane through the intersection of the planes ax + by + cz + d = 0 andlx + my + nz + p = 0 and parallel to the line y=0, z=0


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Find the distance of the point (–1, –5, – 10) from the point of intersection of the line `vec"r" = 2hat"i" - hat"j" + 2hat"k" + lambda(3hat"i" + 4hat"j" + 2hat"k")` and the plane `vec"r" * (hat"i" - hat"j" + hat"k")` = 5


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