Advertisements
Advertisements
Question
Find the shortest distance between lines `vecr = 6hati + 2hatj + 2hatk + lambda(hati - 2hatj + 2hatk)` and `vecr =-4hati - hatk + mu(3hati - 2hatj - 2hatk)`.
Advertisements
Solution
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
RELATED QUESTIONS
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)`
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 whose vector equations are `vecr = (hati + 2hatj + 3hatk) + lambda(hati - 3hatj + 2hatk)` and `vecr = 4hati + 5hatj + 6hatk + mu(2hati + 3hatj + hatk)`.
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:
|
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:
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 ______.
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)`
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.
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 \[\vec a_2\]?
In the Cartesian form for skew lines, what is the denominator of \[\mathbf d\]?
For the Cartesian form of the distance between skew lines, which determinant is the numerator?
For \[\vec a_1=\hat i+\hat j\] and \[\vec a_2=2\hat i+\hat j-\hat k\], what is \[\vec a_2-\vec a_1\]?
For \[\vec b_1=2\hat i-\hat j+\hat k\] and \[\vec b_2=3\hat i-5\hat j+2\hat k\], what is \[\vec b_1\times\vec b_2\]?
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 \[SD\] for intersecting lines?
For skew lines, which expression gives \[SD\]?
