Advertisements
Advertisements
प्रश्न
Find the perpendicular distance of the origin from the plane 6x – 2y + 3z – 7 = 0.
Advertisements
उत्तर
The equation of the plane is
6x – 2y + 3z – 7 = 0
∴ its vector equation is
`bar"r".(6hat"i" - 2hat"j" + 3hat"k")` = 7 ...(1)
where `bar"r" = xhat"i" + yhat"j" + zhat"k"`
∴ `bar"n" = 6hat"i" - 2hat"j" + 3hat"k"` is normal to the plane.
`|bar"n"| = sqrt(6^2 + (-2)^2 + 3^2)`
= `sqrt(49)`
= 7
Unit vector along `bar"n"` is
`hat"n" = bar"n"/|bar"n"|= (6hat"i" - 2hat"j" + 3hat"k")/(7)`
Dividing bothsides of (1) by 7, we get
`bar"r".((6hat"i" - 2hat"j" + 3hat"k")/7) = (7)/(7)`
∴ `bar"r".hat"n"`= 1
Comparing with normal form of equation of the plane `hat"r".hat"n" = p` it follows that length of perpendicular from origin is 1 unit.
Alternative Method :
The equation of the plane is 6x – 2y + 3z – 7 = 0
i.e. `(6/(6^2 + (-2)^2 + 3))x - (2/(sqrt(6^2) + (-2)^2 + 3^2))y + ((3)/(sqrt(6^2 + (-2)^2 + 3^2)))z = 7/(sqrt(6^2 + (-2)^2 + 3)`
i.e. `(6)/(7)x -(2)/(7)y + (3)/(7)z = (7)/(7)` = 1
This is the normal form of the equation of plane.
∴ perpendicular distance of the origin frm the plane is p = 1 unit.
APPEARS IN
संबंधित प्रश्न
Find the co-ordinates of the foot of the perpendicular drawn from the point `2hati - hatj + 5hatk` to the line `barr = (11hati - 2hatj - 8hatk) + λ(10hati - 4hatj - 11hatk).` Also find the length of the perpendicular.
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.
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 coordinates of the foot of the perpendicular drawn from the origin to the plane 2x + 6y – 3z = 63.
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.
Choose correct alternatives :
The length of the perpendicular from (1, 6,3) to the line `x/(1) = (y - 1)/(2) =(z - 2)/(3)`
Choose correct alternatives :
The lines `x/(1) = y/(2) = z/(3) and (x - 1)/(-2) = (y - 2)/(-4) = (z - 3)/(6)` are
Choose correct alternatives :
Equation of X-axis is ______.
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 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 = ______
Find the direction ratios of the normal to the plane 2x + 3y + z = 7
Find direction cosines of the normal to the plane `bar"r"*(3hat"i" + 4hat"k")` = 5
Find the perpendicular distance of origin from the plane 6x − 2y + 3z - 7 = 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 ______.
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 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 ______
Equation of plane parallel to ZX-plane and passing through the point (0, 5, 0) is ______
The equation of the plane through (1, 2, -3) and (2, -2, 1) and parallel to the X-axis is ______
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 ______
A plane which passes through the point (3, 2, 0) and the line `(x - 3)/1 = (y - 6)/5, (z - 4)/4` is ______
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 ______.
Let the line `(x - 2)/3 = (y - 1)/(-5) = (z + 2)/2` lie in the plane x + 3y - αz + β = 0. Then, (α, β) equals ______
The d.r.s of normal to the plane through (1, 0, 0), (0, 1, 0) which makes an angle `pi/4` with plane x + y = 3, are ______.
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 ______.
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`.
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 ______.
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 ______.
Find the equation of the plane containing the lines `(x - 1)/2 = (y + 1)/-1 = z/3` and `x/2 = (y - 2)/-1 = (z + 1)/3`.
Find the equation of plane which is at a distance of 4 units from the origin and which is normal to the vector `2hati - 2hatj + hatk`.
Find the vector equation of the line passing through the point (–2, 1, 4) and perpendicular to the plane `barr*(4hati - 5hatj + 7hatk)` = 15
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 Cartesian equation of a plane through A (7, 8, 6) and parallel to the XY plane is
The direction cosines of the line x - y + 2z = 5 and 3x + y + z = 6 are
