Advertisements
Advertisements
Question
Find the shortest distance between the lines whose vector equations are `vecr = (hati + 2hatj + 3hatk) + lambda(hati - 3hatj + 2hatk)` and `vecr = 4hati + 5hatj + 6hatk + mu(2hati + 3hatj + hatk)`.
Solution
Compare the given equations
`vecr = vec(a_1) + λvec(b_1)` and `vecr = vec(a_2) + µvec(b_2)` respectively, we have `vec(a_1) = hati + 2hatj + 3hatk, vec(b_1) = hati - 3hatj + 2hatk`
`vec(a_2) = 4hati + 5hatj + 6hatk` and `vec(b_2) = 2hati + 3hatj + hatk`
Now, `vec(a_2) - vec(a_1) = 3hati + 3hatj + 3hatk`
And `vec(b_1) xx vec(b_2) = |(hati, hatj, hatk), (1, -3, 2), (2, 3, 1)|`
= `-9hati + 3hatj + 9hatk`
`|vec(b_1) xx vec(b_2)| = 3sqrt19`
∴ `(vec(a_2) - vec(a_1)). (vec(b_1) xx vec(b_2)) = 9`
d = `|((vec(a_2) - vec(a_1)). (vec(b_1) xx vec(b_2)))/|vec(b_1) xx vec(b_2)||`
`= |(-9 xx 3 + 3 xx 3 + 9 xx3)/sqrt((-9)^2 + 3^2 + 9^2)|`
`= |9/ (sqrt(3^2) sqrt(3^2 + 1 + 3^2))|`
= `9/(3sqrt19)`
= `3/sqrt19`
APPEARS IN
RELATED QUESTIONS
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 the following two lines are coplanar:
`(x−a+d)/(α−δ)= (y−a)/α=(z−a−d)/(α+δ) and (x−b+c)/(β−γ)=(y−b)/β=(z−b−c)/(β+γ)`
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 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 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 lines `vecr = 6hati + 2hatj + 2hatk + lambda(hati - 2hatj + 2hatk)` and `vecr =-4hati - hatk + mu(3hati - 2hatj - 2hatk)`.
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
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. |
The most economical speed to run the train 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. |
The fuel cost for the train to travel 500 km at the most economical speed 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. |
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`
The planes `2x - y + 4z` = 5 and `5x - 2.5y + 10z` = 6
Find the shortest distance between the lines, `vecr = 6hati + 2hatj + 2hatk + lambda(hati - 2hatj + 2hatk)` and `vecr = - 4hati - hatk + mu(3hati - 2hatj - 2hatk)`
An insect is crawling along the line `barr = 6hati + 2hatj + 2hatk + λ(hati - 2hatj + 2hatk)` and another insect is crawling along the line `barr = - 4hati - hatk + μ(3hati - 2hatj - 2hatk)`. At what points on the lines should they reach so that the distance between them s the shortest? Find the shortest possible distance between them.
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.
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)`.
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 ______.
Find the distance between the lines:
`vecr = (hati + 2hatj - 4hatk) + λ(2hati + 3hatj + 6hatk)`;
`vecr = (3hati + 3hatj - 5hatk) + μ(4hati + 6hatj + 12hatk)`
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.
