Advertisements
Advertisements
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.
Advertisements
उत्तर
Let M be the foot of perpendicular drawn from the point `P(2hati - hatj + 5hatk)` on the line.
`barr = (11hati - 2hatj - 8hatk) + λ(10hati - 4hatj - 11hatk)`
Let the position vector of the point M be
`(11hati - 2hatj - 8hatk) + λ(10hati - 4hatj - 11hatk)`
= `(11 + 10λ)hati + (-2 - 4λ)hatj + (-8 - 11λ)hatk`.
Then PM = Position vector of M – Position vector of P
= `[(11 + 10λ)hati + (-2 - 4λ)hatj + (-8 - 11λ)hatk] - (2hati -hatj + 5hatk)`
= `(9 + 10λ)hati + (-1 - 4λ)hatj + (-13 - 11λ)hatk`
Since PM is perpendicular to the given line which is parallel to `barb = 10hati - 4hatj - 11hatk`,
PM ⊥ `barb`
∴ PM.`barb` = 0
∴ `[(9 + 10λ)hati + (-1 - 4λ) - 11(-13 - 11λ)hatk].(10hati - 4hatj - 11hatk)` = 0
∴ 10(9 + 10λ) – 4(–1 – 4λ) – 11(13 – 11λ) = 0
∴ 90 + 100λ + 4 + 16λ + 143 + 121λ = 0
∴ 237λ + 237 = 0
∴ λ = – 1
Putting this value of λ, we get the position vector of M as `hati + 2hatj + 3hatk`.
∴ Coordinates of the foot of perpendicular M are (1, 2, 3).
Now, PM = `(hati + 2hatj + 3hatk) - (2hati - hatj + 5hatk)`
= `-hati + 3hatj - 2hatk`
∴ |PM| = `sqrt((-1)^2 + (3)^2 + (-2)^2`
= `sqrt(1 + 9 + 4)`
= `sqrt(14)`
Hence, the coordinates of the foot of perpendicular are (1, 2, 3) and length of perpendicular = `sqrt(14)` units.
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 vector equation of the plane passing through the point having position vector `hati + hatj + hatk` and perpendicular to the vector `4hati + 5hatj + 6hatk`.
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 :
Equation of X-axis 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 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.
The equation of X axis is ______
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 ______
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
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 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 ______.
Equation of the plane passing through A(-2, 2, 2), B(2, -2, -2) and perpendicular to x + 2y - 3z = 7 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 _______.
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, which bisects the line joining the points (1, 2, 3) and (3, 4, 5) at right angles 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 ______.
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 ______
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 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 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 ______.
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?
The equation of the plane passes through the point (2, 5, –3) perpendicular to the plane x + 2y + 2z = 1 and x – 2y + 3z = 4 is ______.
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 ______.
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.
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.
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).
