Advertisements
Advertisements
प्रश्न
Prove that the line through A(0, –1, –1) and B(4, 5, 1) intersects the line through C(3, 9, 4) and D(– 4, 4, 4).
उत्तर
Given points, A(0, –1, –1) and B(4, 5, 1), C(3, 9, 4) and D(– 4, 4, 4).
Cartesian form of equation AB is
`(x - 0)/(4 - 0) = (y + 1)/(5 + 1) = (z + 1)/(1 + 1)`
⇒ `x/4 = (y + 1)/6 = (z + 1)/2`
And it vector form is `vecr = (-hatj - hatk) + lambda(4hati + 6hatj + 2hatk)`
Similarly, equation of Cd is
`(x - 3)/(-4 - 3) = (y - 9)/(4 - 9) = (z - 4)/(4 - 4)`
⇒ `(x - 3)/7 = (y - 9)/(-5) = (z - 4)/0`
And its vector form is `vecr = (3hati + 9hatj + 4hatk) + mu(-7hati - 5hatj)`
Now here `veca_1 = -hatj - hatk, vecb_1 - 4hati + 6hatj + 2hatk`
`veca_2 = 3hati + 9hatj + 4hatk, vecb_2 = -7hati - 5hatj`
Shortest distance between AB and CD
S.D. = `|((veca_2 - veca_1)*(vecb_1 xx vecb_2))/|vecb_1 xx vecb_2||`
`veca_2 - veca_1 = (3hati + 9hatj + 4hatk) - (-hatj - hatk) = 3hati + 10hatj + 5hatk`
`vecb_1 xx vecb_2 = |(hati, hatj, hatk),(4, 6, 2),(-7, -5, 0)|`
= `hati(0 + 10) - hatj(0 + 14) + hatk(-20 + 42)`
= `10hati - 14hatj + 22hatk`
`|vecb_1 xx vecb_2| = sqrt((10)^2 + (-14)^2 + (22)^2)`
= `sqrt(100 + 196 + 484)`
= `sqrt(780)`
∴ S.D. = `((3hati + 10hatj + 5hatk)*(10hati - 14hatj + 22hatk))/sqrt(780)`
= `(30 - 140 + 100)/sqrt(780)`
= 0
Thus, the two lines intersect each other.
APPEARS IN
संबंधित प्रश्न
Find the distance between the pairs of points:
(2, 3, 5) and (4, 3, 1)
Find the distance between the following pairs of points:
(–3, 7, 2) and (2, 4, –1)
Find the distance between the following pairs of points:
(–1, 3, –4) and (1, –3, 4)
Show that the points (–2, 3, 5), (1, 2, 3) and (7, 0, –1) are collinear.
Verify the following:
(0, 7, –10), (1, 6, –6) and (4, 9, –6) are the vertices of an isosceles triangle.
Verify the following:
(0, 7, 10), (–1, 6, 6) and (–4, 9, 6) are the vertices of a right angled triangle.
Find the equation of the set of points P, the sum of whose distances from A (4, 0, 0) and B (–4, 0, 0) is equal to 10.
Find the distance between the following pairs of point:
A(3, 2, –1) and B(–1, –1, –1).
Find the distance between the points P and Q having coordinates (–2, 3, 1) and (2, 1, 2).
Using distance formula prove that the following points are collinear:
A(4, –3, –1), B(5, –7, 6) and C(3, 1, –8)
Determine the points in yz-plane and are equidistant from the points A(1, –1, 0), B(2, 1, 2) and C(3, 2, –1).
Show that the points A(1, 3, 4), B(–1, 6, 10), C(–7, 4, 7) and D(–5, 1, 1) are the vertices of a rhombus.
Prove that the tetrahedron with vertices at the points O(0, 0, 0), A(0, 1, 1), B(1, 0, 1) and C(1, 1, 0) is a regular one.
If the distance between the points P(a, 2, 1) and Q (1, −1, 1) is 5 units, find the value of a.
Write the coordinates of third vertex of a triangle having centroid at the origin and two vertices at (3, −5, 7) and (3, 0, 1).
Find the distance of the point whose position vector is `(2hati + hatj - hatk)` from the plane `vecr * (hati - 2hatj + 4hatk)` = 9
Find the distance of the point (–1, –5, – 10) from the point of intersection of the line `vecr = 2hati - hatj + 2hatk + lambda(3hati + 4hatj + 2hatk)` and the plane `vecr * (hati - hatj + hatk)` = 5.
The distance of a point P(a, b, c) from x-axis is ______.
Find the distance of a point (2, 4, –1) from the line `(x + 5)/1 = (y + 3)/4 = (z - 6)/(-9)`
Find the shortest distance between the lines given by `vecr = (8 + 3lambdahati - (9 + 16lambda)hatj + (10 + 7lambda)hatk` and `vecr = 15hati + 29hatj + 5hatk + mu(3hati + 8hatj - 5hatk)`
Find the equation of the plane through the intersection of the planes `vecr * (hati + 3hatj) - 6` = 0 and `vecr * (3hati + hatj + 4hatk)` = 0, whose perpendicular distance from origin is unity.
Distance of the point (α, β, γ) from y-axis is ______.
The distance of the plane `vecr * (2/4 hati + 3/7 hatj - 6/7hatk)` = 1 from the origin is ______.
If one of the diameters of the circle x2 + y2 – 2x – 6y + 6 = 0 is a chord of another circle 'C' whose center is at (2, 1), then its radius is ______.
