Advertisements
Advertisements
प्रश्न
Show that the lines `(x - 1)/2 = (y - 2)/3 = (z - 3)/4` and `(x - 4)/5 = (y - 1)/2` = z intersect.. Also, find their point of intersection.
Advertisements
उत्तर
We have lines,
`L_1 : (x - 1)/2 = (y - 2)/3 = (z - 3)/4 = lambda`
And `L_2 : (x - 4)/5 = (y - 1)/2 = z = mu`
Any point on the line L1 is (`2lambda + 1, 3lambda + 2, 4lambda + 3)`
Any point on the line L2 is `(5mu + 4, 2mu + 1, mu)`
If line intersect then there exists a value of λ, μ for which
`(2lambda + 1, 3lambda + 2, 4lambda + 3) = (5mu + 4, 2mu + 1, mu)`
⇒ `2lambda + 1 = 5mu + 4, 3 lambda + 2 = 2mu + 1` and `4lambda + 3 = mu`
Solving fisrt two equations we get `lambda = - 1, mu = -1`
These values of `lambda = - 1, mu = - 1` also satisfy the third equation.
Thus lines interest.
Also the point of intersection is `(-1, -1, -1)`.
APPEARS IN
संबंधित प्रश्न
A point is on the x-axis. What are its y-coordinates and z-coordinates?
A point is in the XZ-plane. What can you say about its y-coordinate?
The coordinates of points in the XY-plane are of the form _______.
Find the lengths of the medians of the triangle with vertices A (0, 0, 6), B (0, 4, 0) and (6, 0, 0).
Find the coordinates of a point on y-axis which are at a distance of `5sqrt2` from the point P (3, –2, 5).
A point R with x-coordinate 4 lies on the line segment joining the pointsP (2, –3, 4) and Q (8, 0, 10). Find the coordinates of the point R.
[Hint suppose R divides PQ in the ratio k: 1. The coordinates of the point R are given by `((8k + 2)/(k+1), (-3)/(k+1), (10k + 4)/(k+1))`
A(1, 2, 3), B(0, 4, 1), C(–1, –1, –3) are the vertices of a triangle ABC. Find the point in which the bisector of the angle ∠BAC meets BC.
Find the coordinates of the points which tisect the line segment joining the points P(4, 2, –6) and Q(10, –16, 6).
Given that P(3, 2, –4), Q(5, 4, –6) and R(9, 8, –10) are collinear. Find the ratio in which Qdivides PR.
Find the ratio in which the line segment joining the points (4, 8, 10) and (6, 10, –8) is divided by the yz-plane.
Find the coordinates of a point equidistant from the origin and points A (a, 0, 0), B (0, b, 0) andC(0, 0, c).
P is a point on the line segment joining the points (3, 2, –1) and (6, 2, –2). If x co-ordinate of P is 5, then its y co-ordinate is ______.
The equations of x-axis in space are ______.
The points (1, 2, 3), (–2, 3, 4) and (7, 0, 1) are collinear.
The vector equation of the line passing through the points (3, 5, 4) and (5, 8, 11) is.
Find the vector equation of the line which is parallel to the vector `3hati - 2hatj + 6hatk` and which passes through the point (1 ,–2, 3).
`vec(AB) = 3hati - hatj + hatk` and `vec(CD) = - 3hati + 2hatj + 4hatk` are two vectors. The position vectors of the points A and C are `6hati + 7hatj + 4hatk` and `-9hatj + 2hatk`, respectively. Find the position vector of a point P on the line AB and a point Q on the line CD such that `vec(PQ)` is perpendicular to `vec(AB)` and `vec(CD)` both.
The reflection of the point (α, β, γ) in the xy-plane is ______.
A plane passes through the points (2, 0, 0) (0, 3, 0) and (0, 0, 4). The equation of plane is ______.
The equation of a line, which is parallel to `2hati + hatj + 3hatk` and which passes through the point (5, –2, 4), is `(x - 5)/2 = (y + 2)/(-1) = (z - 4)/3`.
