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)`.
Advertisements
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
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 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 = (hati + 2hatj + 3hatk) + lambda(hati - 3hatj + 2hatk)` and `vecr = 4hati + 5hatj + 6hatk + mu(2hati + 3hatj + hatk)`.
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 `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
|
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)`
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)`
Distance between the planes :-
`2x + 3y + 4z = 4` and `4x + 6y + 8z = 12` is
The planes `2x - y + 4z` = 5 and `5x - 2.5y + 10z` = 6
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 ______.
Find the distance between the lines:
`vecr = (hati + 2hatj - 4hatk) + λ(2hati + 3hatj + 6hatk)`;
`vecr = (3hati + 3hatj - 5hatk) + μ(4hati + 6hatj + 12hatk)`
For parallel lines with common direction vector \[\vec{b}\], which expression gives \[SD\]?
In the Cartesian form for skew lines, what is the denominator of \[\mathbf d\]?
For \[\vec a_1=\hat i+\hat j\] and \[\vec a_2=2\hat i+\hat j-\hat k\], what is \[\vec a_2-\vec a_1\]?
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 \[\vec r=(3\hat i+3\hat j-5\hat k)+\mu(2\hat i+3\hat j+6\hat k)\]?
What is \[SD\] for intersecting lines?
