Advertisements
Advertisements
प्रश्न
Find the shortest distance between the following lines:
`vecr = 3hati + 5hatj + 7hatk + λ(hati - 2hatj + hatk)` and `vecr = (-hati - hatj - hatk) + μ(7hati - 6hatj + hatk)`.
उत्तर
Given lines are: `vecr = 3hati + 5hatj + 7hatk + λ(hati - 2hatj + hatk)`
and `vecr = (-hati - hatj - hatk) + μ(7hati - 6hatj + hatk)`
Let the given lines be `vecr = veca_1 + λvecb_2` and `vecr = veca_2 + λvecb_2`
Shortest distance between two lines
d = `|((veca_2 - veca_1).(vecb_1 xx vecb_2))/|vecb_1 xx vecb_2||`
∴ `veca_2 - veca_1 = (-hati - hatj - hatk) - (3hati + 5hatj + 7hatk)`
= `-4hati - 6hatj - 8hatk`
`vecb_1 xx vecb_2 = |(hati, hatj, hatk),(1, -2, 1),(7, -6, 1)|`
= `hati(-2 + 6) - hatj(1 - 7) + hatk(-6 + 14)`
= `4hati + 6hatj + 8hatk`
∴ `|vecb_1 xx vecb_2| = sqrt(4^2 + 6^2 + 8^2)`
= `sqrt(16 + 36 + 64)`
= `sqrt(116)`
Therefore, d = `|((-4hati - 6hatj - 8hatk).(4hati + 6hatj + 8hatk))/sqrt(116)|`
= `|(-16 - 36 - 64)/sqrt(116)|`
= `|(-116)/sqrt(116)|`
= `sqrt(116)` units
APPEARS IN
संबंधित प्रश्न
If the lines
`(x-1)/-3=(y-2)/(2k)=(z-3)/2 and (x-1)/(3k)=(y-5)/1=(z-6)/-5`
are at right angle then find the value of k
Find the shortest distance between the lines
`bar r = (4 hat i - hat j) + lambda(hat i + 2 hat j - 3 hat k)`
and
`bar r = (hat i - hat j + 2 hat k) + mu(hat i + 4 hat j -5 hat k)`
where λ and μ are parameters
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:
`vecr = (hati+2hatj+hatk) + lambda(hati-hatj+hatk)` and `vecr = 2hati - hatj - hatk + mu(2hati + hatj + 2hatk)`
Find the shortest distance between the lines.
`(x + 1)/7 = (y + 1)/(- 6) = (z + 1)/1` and `(x - 3)/1 = (y - 5)/(- 2) = (z - 7)/1`.
Find the shortest distance between the lines `(x+1)/7=(y+1)/(-6)=(z+1)/1 and (x-3)/1=(y-5)/(-2)=(z-7)/1`
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 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 total cost of the train to travel 500 km at the most economical speed is:
Find the shortest distance between the following lines:
`vecr = (hati + hatj - hatk) + s(2hati + hatj + hatk)`
`vecr = (hati + hatj - 2hatk) + t(4hati + 2hatj + 2hatk)`
Find the equation of line which passes through the point (1, 2, 3) and is parallel to the vector `3hati + 2hatj - 2hatk`
Find the angle between the following pair of lines:- `(x - 2)/ = (y - 1)/5 = (z + 3)/(-3)` and `(x + 2)/(-1) = (y - 4)/8 = (z - 5)/4`
Determine the distance from the origin to the plane in the following case x + y + z = 1
Distance between the planes :-
`2x + 3y + 4z = 4` and `4x + 6y + 8z = 12` is
Find the shortest distance between the lines, `vecr = 6hati + 2hatj + 2hatk + lambda(hati - 2hatj + 2hatk)` and `vecr = - 4hati - hatk + mu(3hati - 2hatj - 2hatk)`
Read the following passage and answer the questions given below.
|
Two motorcycles A and B are running at the speed more than the allowed speed on the roads represented by the lines `vecr = λ(hati + 2hatj - hatk)` and `vecr = (3hati + 3hatj) + μ(2hati + hatj + hatk)` respectively.
|
Based on the above information, answer the following questions:
- Find the shortest distance between the given lines.
- Find the point at which the motorcycles may collide.
The shortest distance between the line y = x and the curve y2 = x – 2 is ______.
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 ______.
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.
