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प्रश्न
Find the direction ratios of the normal to the plane 2x + 3y + z = 7
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उत्तर
The direction ratios of the normal to the plane 2x + 3y + z = 7 are 2, 3, 1.
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संबंधित प्रश्न
Find the length of the perpendicular (2, –3, 1) to the line `(x + 1)/(2) = (y - 3)/(3) = (z + 1)/(-1)`.
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 perpendicular distance of the origin from the plane 6x – 2y + 3z – 7 = 0.
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.
Find the vector equation of the plane passing through the point having position vector `hati + hatj + hatk` and perpendicular to the vector `4hati + 5hatj + 6hatk`.
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 :
Equation of X-axis 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 coordinates of the foot of the perpendicular drawn from the origin to the plane 2x + 3y + 6z = 49.
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.
If the foot of the perpendicular drawn from the origin to the plane is (4, −2, -5), then the equation of the plane is ______
If the normal to the plane has direction ratios 2, −1, 2 and it’s perpendicular distance from origin is 6, find its equation
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
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 line of intersection of the planes 2x - y - 4 = 0 and y + 2z - 4 = 0 and passing through the point (1, 1, 0) is ______
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 = ______.
The equation of the plane through (1, 2, -3) and (2, -2, 1) and parallel to the X-axis 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 ______
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 ______
The distance of the point (1, 0, 2) from the point of intersection of the line `(x - 2)/3 = (y + 1)/4 = (z - 2)/12` and the plane x - y + z = 16, is ______
Find the coordinates of the foot of the perpendicular drawn from the origin to the plane 3y + 5 = 0.
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 the plane x - 3y + 5z = d passes through the point (1, 2, 4), then the lengths of intercepts cut by it on the axes of X, Y, Z are respectively ______
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 ______.
If plane x + ay + z = 4 has equal intercepts on axes, then 'a' is equal to ______.
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 P be a plane Ix + my + nz = 0 containing the line, `(1 - x)/1 = ("y" + 4)/2 = ("z" + 2)/3`. If plane P divides the line segment AB joining points A(–3, –6, 1) and B(2, 4, –3) in ratio k:1 then the value of k is equal to ______.
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 ______.
What will be the equation of plane passing through a point (1, 4, – 2) and parallel to the given plane – 2x + y – 3z = 9?
If the foot of the perpendicular drawn from the origin to the plane is (4, –2, 5), then the equation of the plane is ______.
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 point of intersection of the line `(x + 1)/2 = (y - 1)/3 = (z - 2)/1` with the plane x + 2y – z = 6.
A mobile tower is situated at the top of a hill. Consider the surface on which the tower stands as a plane having points A(1, 0, 2), B(3, –1, 1) and C(1, 2, 1) on it. The mobile tower is tied with three cables from the points A, B and C such that it stands vertically on the ground. The top of the tower is at point P(2, 3, 1) as shown in the figure below. The foot of the perpendicular from the point P on the plane is at the point `Q(43/29, 77/29, 9/29)`.
Answer the following questions.
- Find the equation of the plane containing the points A, B and C.
- Find the equation of the line PQ.
- Calculate the height of the tower.
Find the equation of the plane containing the line `x/(-2) = (y - 1)/3 = (1 - z)/1` and the point (–1, 0, 2).
The Cartesian equation of a plane through A (7, 8, 6) and parallel to the XY plane is
