Advertisements
Advertisements
प्रश्न
Find the vector equation of a plane at a distance 6 units from the origin and to which vector `2hat"i" - hat"j" + 2hat"k"` is normal
उत्तर
Let `bar"n" = 2hat"i" - hat"j" + 2hat"k"`
∴ `hat"n"` is the unit vector along normal
∴ `hat"n" = (bar"n")/|bar"n"|`
= `(2hat"i" - hat"j" + 2hat"k")/sqrt(2^2 + (-1)^2 2 ^2)`
= `(2hat"i" - hat"j" + 2hat"k")/sqrt(4 + 1 + 4)`
= `(2hat"i" - hat"j" + 2hat"k")/3`
and p = 6
Vector equation of plane is `bar"r"*hat"n"` = p
∴ `bar"r"*((2hat"i" - hat"j" + 2hat"k"))/3` = 6
∴ `bar"r"*(2hat"i" - hat"j" + 2hat"k")` = 18
APPEARS IN
संबंधित प्रश्न
Find the length of the perpendicular (2, –3, 1) to the line `(x + 1)/(2) = (y - 3)/(3) = (z + 1)/(-1)`.
Find the perpendicular distance of the point (1, 0, 0) from the line `(x - 1)/(2) = (y + 1)/(-3) = (z + 10)/(8)` Also find the co-ordinates of the foot of the perpendicular.
A(1, 0, 4), B(0, -11, 13), C(2, -3, 1) are three points and D is the foot of the perpendicular from A to BC. Find the co-ordinates of D.
Find the vector equation of a plane which is at 42 unit distance from the origin and which is normal to the vector `2hati + hatj - 2hatk`.
Find the perpendicular distance of the origin from the plane 6x – 2y + 3z – 7 = 0.
Find the coordinates of the foot of the perpendicular drawn from the origin to the plane 2x + 6y – 3z = 63.
Show that the line `bar"r" = (2hat"j" - 3hat"k") + lambda(hat"i" + 2hat"j" + 3hat"k") and bar"r" = (2hat"i" + 6hat"j" + 3hat"k") + mu(2hat"i" + 3hat"j" + 4hat"k")` are coplanar. Find the equation of the plane determined by them.
Find the co-ordinates of the foot of the perpendicular drawn from the point (0, 2, 3) to the line `(x + 3)/(5) = (y - 1)/(2) = (z + 4)/(3)`.
Choose correct alternatives :
The equation of the plane passing through (2, -1, 3) and making equal intercepts on the coordinate axes is
Choose correct alternatives :
The equation of the plane passing through the points (1, −1, 1), (3, 2, 4) and parallel to the Y-axis is ______
Solve the following :
Find the perpendicular distance of the origin from the plane 6x + 2y + 3z - 7 = 0
If the foot of the perpendicular drawn from the origin to the plane is (4, −2, -5), then the equation of the plane is ______
Find direction cosines of the normal to the plane `bar"r"*(3hat"i" + 4hat"k")` = 5
If the normal to the plane has direction ratios 2, −1, 2 and it’s perpendicular distance from origin is 6, find its equation
Find the equation of the plane passing through the point (7, 8, 6) and parallel to the plane `bar"r"*(6hat"i" + 8hat"j" + 7hat"k")` = 0
Find the vector equation of the plane which bisects the segment joining A(2, 3, 6) and B(4, 3, −2) at right angles
If z1 and z2 are z-coordinates of the points of trisection of the segment joining the points A (2, 1, 4), B (–1, 3, 6) then z1 + z2 = ______.
The equation of a plane containing the point (1, - 1, 2) and perpendicular to the planes 2x + 3y - 2z = 5 and x + 2y - 3z = 8 is ______.
The intercepts of the plane 3x - 4y + 6z = 48 on the co-ordinate axes are ______
Equations of planes parallel to the plane x - 2y + 2z + 4 = 0 which are at a distance of one unit from the point (1, 2, 3) are _______.
The equation of the plane, which bisects the line joining the points (1, 2, 3) and (3, 4, 5) at right angles is ______
A plane which passes through the point (3, 2, 0) and the line `(x - 3)/1 = (y - 6)/5, (z - 4)/4` is ______
If the plane passing through the points (1, 2, 3), (2, 3, 1) and (3, 1, 2) is ax + by + cz = d then a + 2b + 3c = ______.
The equation of the plane passing through the intersection of the planes x + 2y + 3z + 4 = 0 and 4x + 3y + 2z + 1 = 0 and the origin is ______.
If plane x + ay + z = 4 has equal intercepts on axes, then 'a' is equal to ______.
If the line `(x + 1)/2 = (y - 5)/3 = (z - "p")/6` lies in the plane 3x – 14y + 6z + 49 = 0, then the value of p is ______.
The equation of the 1 plane passing through the points (1, –1, 1), (3, 2, 4) and parallel to Y-axis is ______.
Let Q be the mirror image of the point P(1, 2, 1) with respect to the plane x + 2y + 2z = 16. Let T be a plane passing through the point Q and contains the line `vecr = -hatk + λ(hati + hatj + 2hatk)`, λ ∈ R. Then, which of the following points lies on T?
If A and B are foot of perpendicular drawn from point Q(a, b, c) to the planes yz and zx, then equation of plane through the points A, B and O is ______.
If the foot of the perpendicular drawn from the origin to the plane is (4, –2, 5), then the equation of the plane is ______.
Find the equation of the plane which contains the line of intersection of the planes x + 2y + 4z = 4 and 2x – 3y – z = 9 and which is perpendicular to the plane 4x – 3y + 5z = 10.
The perpendicular distance of the plane `bar r. (3 hat i + 4 hat j + 12 hat k) = 78` from the origin is ______.
