Advertisements
Advertisements
प्रश्न
Find the length of the perpendicular (2, –3, 1) to the line `(x + 1)/(2) = (y - 3)/(3) = (z + 1)/(-1)`.
Advertisements
उत्तर १
Let PM be the perpendicular drawn from the point P(2, –3, 1) to the line `(x + 1)/(2) = (y - 3)/(3) = (z + 1)/(-1) = λ` ...(Say)
The coordinates of any point on the line are given by
x = – 1 + 2λ, y = 3 + 3λ, z = – 1 – λ
Let the coordinates of M be
(–1 + 2λ, 3 + 3λ, –1 – λ) ...(1)
The direction ratios of PM are
–1 + 2λ – 2, 3 + 3λ + 3, –1 – λ –1
i.e. 2λ – 3, 3λ + 6, – λ – 2
The direction ratios of the given line are 2, 3, –1.
Since PM is perpendicular to the given line, we get
2(2λ – 3) + 3(3λ + 6) – 1(– λ – 2) = 0
∴ 4λ – 6 + 9λ + 18 + λ + 2 = 0
∴ 14λ + 14 = 0
∴ λ = – 1.
Put λ = – 1 in (1), the coordinates of M are
(– 1 – 2, 3 – 3, – 1 + 1) i.e. (– 3, 0, 0).
∴ length of perpendicular from P to the given line
= PM
= `sqrt((- 3 - 2)^2 + (0 + 3)^2 + (0 - 1)^2)`
= `sqrt((25 + 9 + 1)`
= `sqrt(35) "units"`.
उत्तर २
We know that the perpendicular distance from the point
`"P"|barα| "to the line" bar"r" = bar"a" + λbar"b"` is given by
`sqrt(|barα - bar"a"|^2 - [((barα - bara).barb)/[|bar"b"|)]^2` ...(1)
Here, `barα = 2hat"i" - 3hat"j" + hat"k", bar"a" = -hat"i" + 3hat"j" - hat"k", bar"b" = 2hat"i" + 3hat"j" - hat"k"`
∴ `barα - bar"a" = (2hat"i" - 3hat"j" + hat"k") - (-hat"i" + 3hat"j" - hat"k")`
= `3hat"i" - 6hat"j" + 2hat"k"`
∴ `|barα - bar"a"|^2` = 32 + (– 6)2 + 22 = 9 + 36 + 4 = 49
Also, `(barα - bar"a").bar"b" = (3hat"i" - 6hat"j" + 2hat"k").(2hat"i" + 3hat"j" - hat"k")`
= (3)(2) + (– 6)(3) + (2)(– 1)
= 6 – 18 – 2
= – 14
`|bar"b"| = sqrt(2^2 + 3^2 + (-1)^2) = sqrt(14)`
Substitutng these values in (1), we get
length of perpendicular from P to given line
= PM
= `sqrt(49 - ((-14)/sqrt(14))^2`
= `sqrt(49 - 14)`
= `sqrt(35) "units"`.
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.
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.
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:
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 ______.
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 :
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 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 planes 2x – my + z = 3 and 4x – y + 2z = 5 are parallel then m = ______
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 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 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
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
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 ______
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 ______.
The intercepts of the plane 3x - 4y + 6z = 48 on the co-ordinate axes 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 = ______.
XY-plane divides the line joining the points A(2, 3, -5) and B(1, -2, -3) in the ratio ______
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 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 ______
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 ______.
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 ______.
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?
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 ______.
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`.
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 perpendicular distance of the plane `bar r. (3 hat i + 4 hat j + 12 hat k) = 78` from the origin is ______.
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
The direction cosines of the line x - y + 2z = 5 and 3x + y + z = 6 are
