Advertisements
Advertisements
प्रश्न
Solve the following :
Reduce the equation `bar"r".(6hat"i" + 8hat"j" + 24hat"k")` = 13 normal form and hence find
(i) the length of the perpendicular from the origin to the plane.
(ii) direction cosines of the normal.
Advertisements
उत्तर
The normal form of equation of a plane is `bar"r".hat"n" = p` where `hat"n"` is unit vector along the normal and p is the length of perpendicular drawn from origin to the plane.
Given pane is `bar"r".(6hat"i" + 8hat"j" + 24hat"k")` = 13 ...(1)
`bar"n" = 6hat"i" + 8hat"j" + 24hat"k"` is normal to the plane
∴ `|bar"n"| = sqrt(6^2 + 8^2 +24^2) = sqrt(36 + 64 + 576) = sqrt(676) = 26`
Dividing both sides of (1) by 26, get
`bar"r".(6/26hat"i" + 8/26hat"j" + 24/26hat"k") = (13)/(26)`
`bar"r".(3/13hat"i" + 4/13hat"j" + 12/13hat"k") = 1/2`
This is the normal form of the equation of plane.
Comparing with `bar"r".hat"n" = p`,
(i) the length of the perpendicular from the origin to plane is `(1)/(2)`.
(ii) direction cosines of the normal are `(3)/(13),(4)/(13),(12)/(13)`.
APPEARS IN
संबंधित प्रश्न
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.
Reduce the equation `bar"r".(3hat"i" + 4hat"j" + 12hat"k")` to normal form and hence find
(i) the length of the perpendicular from the origin to the plane
(ii) direction cosines of the normal.
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.
Choose correct alternatives:
If the line `x/(3) = y/(4)` = z is perpendicular to the line `(x - 1)/k = (y + 2)/(3) = (z - 3)/(k - 1)`, then the value of k is ______.
The perpendicular distance of the plane 2x + 3y – z = k from the origin is `sqrt(14)` units, the value of k is ______.
Choose correct alternatives :
The direction cosines of the normal to the plane 2x – y + 2z = 3 are ______
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 ______
Choose correct alternatives :
The foot of perpendicular drawn from the point (0,0,0) to the plane is (4, -2, -5) then the equation of the plane is
Solve the following :
Find the perpendicular distance of the origin from the plane 6x + 2y + 3z - 7 = 0
Solve the following :
Find the coordinates of the foot of the perpendicular drawn from the origin to the plane 2x + 3y + 6z = 49.
If the planes 2x – my + z = 3 and 4x – y + 2z = 5 are parallel then m = ______
The coordinates of the foot of perpendicular drawn from the origin to the plane 2x + y − 2z = 18 are ______
Find the direction ratios of the normal to the plane 2x + 3y + z = 7
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
Show that the lines `(x + 1)/(-10) = (y + 3)/(-1) = (z - 4)/(1)` and `(x + 10)/(-1) = (y + 1)/(-3) = (z - 1)/4` intersect each other.also find the coordinates of the point of intersection
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 ______.
If the line `(x - 3)/2 = (y + 2)/-1 = (z + 4)/3` lies in the plane lx + my - z = 9, then l2 + m2 is equal to ______
If 0 ≤ x < 2π, then the number of real values of x, which satisfy the equation cos x + cos 2x + cos 3x + cos 4x = 0, is ______
The equation of the plane passing through the point (– 1, 2, 1) and perpendicular to the line joining the points (– 3, 1, 2) and (2, 3, 4) is ______.
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 _______.
If line `(2x - 4)/lambda = ("y" - 1)/2 = ("z" - 3)/1` and `(x - 1)/1 = (3"y" - 1)/lambda = ("z" - 2)/1` are perpendicular to each other then λ = ______.
Equation of the plane perpendicular to the line `x/1 = y/2 = z/3` and passing through the point (2, 3, 4) is ______
The equation of the plane through the point (2, -1, -3) and parallel to the lines `(x - 1)/3 = (y + 2)/2 = z/(-4)` and `x/2 = (y - 1)/(-3) = (z - 2)/2` 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 points (1, –2, 1), (2, –1, –3) and (0, 1, 5) is ______.
The equation of the plane passing through a point having position vector`-2hat"i" + 7hat"j" + 5hat"k"` and parallel to the vectors `4hat"i" - hat"j" + 3hat"k"` and `hat"i" + hat"j" + hat"k"` is ______.
The equation of the 1 plane passing through the points (1, –1, 1), (3, 2, 4) and parallel to Y-axis is ______.
Find the vector equation of the plane passing through the point A(–1, 2, –5) and parallel to the vectors `4hati - hatj + 3hatk` and `hati + hatj - hatk`.
If the mirror image of the point (2, 4, 7) in the plane 3x – y + 4z = 2 is (a, b, c), then 2a + b + 2c is equal to ______.
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?
Let P be a plane passing through the points (1, 0, 1), (1, –2, 1) and (0, 1, –2). Let a vector `vec"a" = αhat"i" + βhat"j" + γhat"k"` be such that `veca` is parallel to the plane P, perpendicular to `(hat"i"+2hat"j"+3hat"k")`and `vec"a".(hat"i" + hat"j" + 2hat"j")` = 2, then (α – β + γ)2 equals ______.
The equation of the plane through the line x + y + z + 3 = 0 = 2x – y + 3z + 1 and parallel to the line `x/1 = y/2 = z/3`, is ______.
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 ______.
Reduce the equation `barr*(3hati - 4hatj + 12hatk)` = 3 to the normal form and hence find the length of perpendicular from the origin to the plane.
The coordinates of the foot of the perpendicular from the point P(1, 0, 0) in the line `(x - 1)/2 = (y + 1)/-3 = (z + 10)/8` are ______.
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.
Find the equation of the plane containing the line `x/(-2) = (y - 1)/3 = (1 - z)/1` and the point (–1, 0, 2).
