Advertisements
Advertisements
प्रश्न
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)`
उत्तर
We have `vecr = (8 + 3lambdahati - (9 + 16lambda)hatj + (10 + 7lambda)hatk`
= `8hati + 9hatj + 10hatk + lambda(3hati - 16hatj + 7hatk)`
⇒ `veca_1 = 8hati - 9hatj + 10hatk` and `vecb_1 = 3hati - 16hatj + 7hatk` .....(i)
Aslo `vecr = 15hati + 29hatj + 5hatk + mu(3hati + 8hatj - 5hatk)`
⇒ `veca_2 = 15hati + 29hatj + 5hatk` and `vecb_2 = 3hati + 8hatj - 5hatk` .....(ii)
Now, shortest distance between two lines is given by
`|((vecb_1 xx vecb_2) * (veca_2 - veca_2))/|vecb_1 xx vecb_2||`
∴ `vecb_1 xx vecb_2 = |(hati, hatj, hatk),(3, -16, 7),(3, 8, -5)|`
= `hati(80 - 56) - hatj(-15 - 21) + hatk(24 + 48)`
= `24hati + 36hatj + 72hatk`
⇒ `|vecb_1 xx vecb_2| = sqrt(24^2 + 36^2 + 72^2)`
= `12sqrt(2^2 + 3^2 + 6^2)`
= 84
Now `(veca_2 - veca_1) = (15 - 8)hati + (29 + 9)hatj + (5 - 10)hatk`
= `7hati + 38hatj - 5hatk`
∴ Shortest distance = `|((24hati + 36hatj + 72hatk) * (7hati + 38hatj - 5hatk))/84|`
= `|((2hati + 3hatj + 6hatk) * (7hati + 38hatj - 5hatk))/7|`
= `|(14 + 114 - 30)/7|`
= 14
APPEARS IN
संबंधित प्रश्न
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)
Verify the following:
(–1, 2, 1), (1, –2, 5), (4, –7, 8) and (2, –3, 4) are the vertices of a parallelogram.
Find the equation of the set of points which are equidistant from the points (1, 2, 3) and (3, 2, –1).
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 points P and Q having coordinates (–2, 3, 1) and (2, 1, 2).
Using distance formula prove that the following points are collinear:
A(3, –5, 1), B(–1, 0, 8) and C(7, –10, –6)
Show that the points (0, 7, 10), (–1, 6, 6) and (–4, 9, 6) are the vertices of an isosceles right-angled triangle.
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.
Show that the points (3, 2, 2), (–1, 4, 2), (0, 5, 6), (2, 1, 2) lie on a sphere whose centre is (1, 3, 4). Find also its radius.
Find the centroid of a triangle, mid-points of whose sides are (1, 2, –3), (3, 0, 1) and (–1, 1, –4).
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 (– 2, 4, – 5) from the line `(x + 3)/3 = (y - 4)/5 = (z + 8)/6`
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 angle between the lines `vecr = 3hati - 2hatj + 6hatk + lambda(2hati + hatj + 2hatk)` and `vecr = (2hatj - 5hatk) + mu(6hati + 3hatj + 2hatk)`
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).
Find the equation of a plane which is at a distance `3sqrt(3)` units from origin and the normal to which is equally inclined to coordinate axis
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 ______.
The points A(5, –1, 1); B(7, –4, 7); C(1, –6, 10) and D(–1, –3, 4) are vertices of a ______.
