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:
(–1, 3, –4) and (1, –3, 4)
Verify the following:
(0, 7, 10), (–1, 6, 6) and (–4, 9, 6) are the vertices of a right angled triangle.
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 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 points:
P(1, –1, 0) and Q(2, 1, 2)
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)
Using distance formula prove that the following points are collinear:
P(0, 7, –7), Q(1, 4, –5) and R(–1, 10, –9)
Using distance formula prove that the following points are collinear:
A(3, –5, 1), B(–1, 0, 8) and C(7, –10, –6)
Determine the points in xy-plan are equidistant from the points A(1, –1, 0), B(2, 1, 2) and C(3, 2, –1).
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 (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.
Find the centroid of a triangle, mid-points of whose sides are (1, 2, –3), (3, 0, 1) and (–1, 1, –4).
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).
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)`
Find the distance of a point (2, 4, –1) from the line `(x + 5)/1 = (y + 3)/4 = (z - 6)/(-9)`
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 ______.
