Advertisements
Advertisements
प्रश्न
Find the shortest distance between lines `vecr = 6hati + 2hatj + 2hatk + lambda(hati - 2hatj + 2hatk)` and `vecr =-4hati - hatk + mu(3hati - 2hatj - 2hatk)`.
Advertisements
उत्तर
If two points, a2 on the line and b1 on the line, and 22 is their direction, then the shortest distance between them is
If two points `vec(a_1), vec(a_2)` lie on a line and `vec(b_1)` and `vec(b_2`
their direction then minimum distance between them = `|((vec(a_1) - vec(a_2)). (vec(b_1) xx vec(b_2)))/|vec(b_1) xx vec(b_2)||`
Given lines, `vecr = 6hati + 2hatj + 2hatk + λ(hati - 2hatj+ 2hatk)`
And `vecr = -4hati - hatk + µ(3hati - 2hatj+ 2hatk)`
If `vec(a_1) = 6hati + 2hatj + 2hatk`, `vec(a_2) = -4hati - hatk`
`vec(b_1) = hati - 2hatj + 2hatk`, `vec(b_2) = 3hati - 2hatk -2hatk`
`vec(a_1) - vec(a_2) = (6hati + 2hatj + 2hatk) - (-4hati - hatk)`
= `10hati + 2hatj + 3hatk`
`vec(b_1) xx vec(b_2) = |(hati, hatj, hatk), (1, -2, 2), (3, -2, -2)|`
= `8hati + 8hatj + 4hatk`
`|vec(b_1) xx vec(b_2)|`
= `sqrt(8^2 + 8^2 + 4^2)`
= 12
∴ S.D. = `((vec(a_1) - vec(a_2)). (vec(b_1) xx vec(b_2)))/|vec(b_1) xx vec(b_2)|`
= `((10hati + 2hatj + 3hatk).(8hati + 8hatj + 4hatk))/12`
= `(80 + 16 + 12)/12`
= `108/12`
= 9
APPEARS IN
संबंधित प्रश्न
Show that lines:
`vecr=hati+hatj+hatk+lambda(hati-hat+hatk)`
`vecr=4hatj+2hatk+mu(2hati-hatj+3hatk)` are coplanar
Also, find the equation of the plane containing these lines.
Find the distance between the planes 2x - y + 2z = 5 and 5x - 2.5y + 5z = 20
Find the shortest distance between the lines whose vector equations are `vecr = (1-t)hati + (t - 2)hatj + (3 -2t)hatk` and `vecr = (s+1)hati + (2s + 1)hatk`.
Find the shortest distance between the lines `vecr = (4hati - hatj) + lambda(hati+2hatj-3hatk)` and `vecr = (hati - hatj + 2hatk) + mu(2hati + 4hatj - 5hatk)`
Find the shortest distance between the lines `vec r = hat i + 2hat j + 3 hat k + lambda(2 hat i + 3hatj + 4hatk)` and `vec r = 2hat i + 4 hat j + 5 hat k + mu (4hat i + 6 hat j + 8 hat k)`
Find the shortest distance between the lines
Find the shortest distance between the lines
Find the shortest distance between the lines
Find the shortest distance between the lines given by `vec"r" = (8 + 3lambdahat"i" - (9 + 16lambda)hat"j" + (10 + 7lambda)hat"k"` and `vec"r" = 15hat"i" + 29hat"j" + 5hat"k" + mu(3hat"i" + 8hat"j" - 5hat"k")`
|
The fuel cost per hour for running a train is proportional to the square of the speed it generates in km per hour. If the fuel costs ₹ 48 per hour at a speed of 16 km per hour and the fixed charges to run the train amount to ₹ 1200 per hour. Assume the speed of the train as v km/h. |
Given that the fuel cost per hour is k times the square of the speed the train generates in km/h, the value of k is:
|
The fuel cost per hour for running a train is proportional to the square of the speed it generates in km per hour. If the fuel costs ₹ 48 per hour at a speed of 16 km per hour and the fixed charges to run the train amount to ₹ 1200 per hour. Assume the speed of the train as v km/h. |
If the train has travelled a distance of 500 km, then the total cost of running the train is given by the function:
|
The fuel cost per hour for running a train is proportional to the square of the speed it generates in km per hour. If the fuel costs ₹ 48 per hour at a speed of 16 km per hour and the fixed charges to run the train amount to ₹ 1200 per hour. Assume the speed of the train as v km/h. |
The most economical speed to run the train is:
What will be the shortest distance between the lines, `vecr = (hati + 2hatj + hatk) + lambda(hati - hatj + hatk)` and `vecr = (2hati - hatj - hatk) + mu(2hati + hatj + 2hatk)`
Determine the distance from the origin to the plane in the following case x + y + z = 1
If the shortest distance between the lines `vecr_1 = αhati + 2hatj + 2hatk + λ(hati - 2hatj + 2hatk)`, λ∈R, α > 0 `vecr_2 = - 4hati - hatk + μ(3hati - 2hatj - 2hatk)`, μ∈R is 9, then α is equal to ______.
The largest value of a, for which the perpendicular distance of the plane containing the lines `vec"r" = (hat"i" + hat"j") + λ(hat"i" + "a"hat"j" - hat"k")` and `vec"r" = (hat"i" + hat"j") + μ(-hat"i" + hat"j" - "a"hat"k")` from the point (2, 1, 4) is `sqrt(3)`, is ______.
If the shortest distance between the lines `(x - 1)/2 = (y - 2)/3 = (z - 3)/λ` and `(x - 2)/1 = (y - 4)/4 = (z - 5)/5` is `1/sqrt(3)`, then the sum of all possible values of λ is ______.
The shortest distance between the z-axis and the line x + y + 2z – 3 = 0 = 2x + 3y + 4z – 4, is ______.
The lines `vecr = hati + hatj - hatk + λ(2hati + 3hatj - 6hatk)` and `vecr = 2hati - hatj - hatk + μ(6hati + 9hatj - 18hatk)`; (where λ and μ are scalars) are ______.
An aeroplane is flying along the line `vecr = λ(hati - hatj + hatk)`; where 'λ' is a scalar and another aeroplane is flying along the line `vecr = hati - hatj + μ(-2hatj + hatk)`; where 'μ' is a scalar. At what points on the lines should they reach, so that the distance between them is the shortest? Find the shortest possible distance between them.
Show that the line whose vector equation is `vecr = (2hati - 2hatj + 3hatk) + λ(hati - hatj + 4hatk)` is parallel to the plane whose vector equation is `vecr.(hati + 5hatj + hatk) = 5`. Also find the distance between them.
What does the concept of shortest distance help in finding when two lines do not meet directly?
For parallel lines with common direction vector \[\vec{b}\], which expression gives \[SD\]?
Given \[\vec r=\vec a_1+\lambda\vec b_1\] and \[\vec r=\vec a_2+\mu\vec b_2\], what are \[\vec a_1\] and \[\vec a_2\]?
Given \[\vec r=\vec a_1+\lambda\vec b_1\] and \[\vec r=\vec a_2+\mu\vec b_2\], what are \[\vec b_1\] and \[\vec b_2\]?
Which vector form gives the distance \[\mathbf d\] between skew lines?
In the Cartesian form for skew lines, what is the denominator of \[\mathbf d\]?
For the parallel lines with \[\vec a_2-\vec a_1=2\hat i+\hat j-\hat k\] and \[\vec b=2\hat i+3\hat j+6\hat k\], what is \[\vec b\times(\vec a_2-\vec a_1)\]?
What is the distance between \[\vec r=(\hat i+2\hat j-4\hat k)+\lambda(2\hat i+3\hat j+6\hat k)\] and \[\vec r=(3\hat i+3\hat j-5\hat k)+\mu(2\hat i+3\hat j+6\hat k)\]?
For skew lines, which expression gives \[SD\]?
