Advertisements
Advertisements
प्रश्न
Find the Cartesian equation of the following planes:
`vecr.(2hati + 3hatj-4hatk) = 1`
Advertisements
उत्तर
`vecr.(2hati + 3hatj-4hatk)` = 1 ....(1)
For any arbitrary point P (x, y, z) on the plane, position vector `vecr` is given by,`vecr = xhati + yhatj - zhatk`
Substituting the value of `vecr` in equation (1), we obtain
This is the Cartesian equation of the plane.
APPEARS IN
संबंधित प्रश्न
Find the vector equation of the plane passing through a point having position vector `3 hat i- 2 hat j + hat k` and perpendicular to the vector `4 hat i + 3 hat j + 2 hat k`
Write the vector equation of the plane, passing through the point (a, b, c) and parallel to the plane `vec r.(hati+hatj+hatk)=2`
Find the vector equation of the plane which contains the line of intersection of the planes `vecr (hati+2hatj+3hatk)-4=0` and `vec r (2hati+hatj-hatk)+5=0` which is perpendicular to the plane.`vecr(5hati+3hatj-6hatk)+8=0`
Find the vector equation of the plane passing through three points with position vectors ` hati+hatj-2hatk , 2hati-hatj+hatk and hati+2hatj+hatk` . Also find the coordinates of the point of intersection of this plane and the line `vecr=3hati-hatj-hatk lambda +(2hati-2hatj+hatk)`
Find the vector equation of a plane which is at a distance of 5 units from the origin and its normal vector is `2hati-3hatj+6hatk`
The x-coordinate of a point of the line joining the points P(2,2,1) and Q(5,1,-2) is 4. Find its z-coordinate
Find the Cartesian equation of the following planes:
`vecr.(hati + hatj-hatk) = 2`
Find the Cartesian equation of the following planes:
`vecr.[(s-2t)hati + (3 - t)hatj + (2s + t)hatk] = 15`
In the following cases, find the coordinates of the foot of the perpendicular drawn from the origin.
3y + 4z – 6 = 0
In the following cases, find the coordinates of the foot of the perpendicular drawn from the origin.
x + y + z = 1
Find the vector and Cartesian equation of the planes that passes through the point (1, 0, −2) and the normal to the plane is `hati + hatj - hatk`
Find the vector and Cartesian equation of the planes that passes through the point (1, 4, 6) and the normal vector to the plane is `hati -2hatj + hatk`
Find the equation of the plane through the line of intersection of `vecr*(2hati-3hatj + 4hatk) = 1`and `vecr*(veci - hatj) + 4 =0`and perpendicular to the plane `vecr*(2hati - hatj + hatk) + 8 = 0`. Hence find whether the plane thus obtained contains the line x − 1 = 2y − 4 = 3z − 12.
The Cartesian equation of the line is 2x - 3 = 3y + 1 = 5 - 6z. Find the vector equation of a line passing through (7, –5, 0) and parallel to the given line.
Find the image of a point having the position vector: `3hati - 2hatj + hat k` in the plane `vec r.(3hati - hat j + 4hatk) = 2`
Find the vector and Cartesian equations of the line passing through (1, 2, 3) and parallel to the planes \[\vec{r} \cdot \left( \hat{i} - \hat{j} + 2 \hat{k} \right) = 5 \text{ and } \vec{r} \cdot \left( 3 \hat{i} + \hat{j} + 2 \hat{k} \right) = 6\]
Find the vector equation of the line passing through (1, 2, 3) and parallel to the planes \[\vec{r} \cdot \left( \hat{i} - \hat{j} + 2 \hat{k} \right) = 5 \text{ and } \vec{r} \cdot \left( 3 \hat{i} + \hat{j} + \hat{k} \right) = 6 .\]
Find the equation of the plane passing through the intersection of the planes `vec(r) .(hat(i) + hat(j) + hat(k)) = 1"and" vec(r) . (2 hat(i) + 3hat(j) - hat(k)) +4 = 0 `and parallel to x-axis. Hence, find the distance of the plane from x-axis.
Find the Cartesian equation of the plane, passing through the line of intersection of the planes `vecr. (2hati + 3hatj - 4hatk) + 5 = 0`and `vecr. (hati - 5hatj + 7hatk) + 2 = 0` intersecting the y-axis at (0, 3).
Find the vector and cartesian equation of the plane passing through the point (2, 5, - 3), (-2, -3, 5) and (5, 3, -3). Also, find the point of intersection of this plane with the line passing through points (3, 1, 5) and (-1, -3, -1).
Vector equation of a line which passes through a point (3, 4, 5) and parallels to the vector `2hati + 2hatj - 3hatk`.
Find the vector and Cartesian equations of the plane passing through the points (2, 2 –1), (3, 4, 2) and (7, 0, 6). Also find the vector equation of a plane passing through (4, 3, 1) and parallel to the plane obtained above.
The vector equation of the line `(x - 5)/3 = (y + 4)/7 = (z - 6)/2` is ______.
The vector equation of the line through the points (3, 4, –7) and (1, –1, 6) is ______.
