Advertisements
Advertisements
Question
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
Advertisements
Solution
Given equations of the line are:
Let `bara` and `bar b` be vectors in the direction of lines `(x-1)/-3=(y-2)/(2k)=(z-3)/2` and `(x-1)/(3k)=(y-5)/1=(z-6)/-5 " respectively"`
`therefore bar a=-3hati+2khatj+2hatk and bar b=3khati+hatj-5hatk`
`bar a.bar b=-9k+2k-10=-7k-10`
Given lines are at right angle
`theta=90^@`
`costheta=(bar a.bar b)/(|bara|.|barb|)`
`bara.barb=0`
`-7k-10=0`
`k=-10/7`
APPEARS IN
RELATED QUESTIONS
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 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
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. |
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:
|
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 equation of line which passes through the point (1, 2, 3) and is parallel to the vector `3hati + 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)`
Determine the distance from the origin to the plane in the following case x + y + z = 1
The planes `2x - y + 4z` = 5 and `5x - 2.5y + 10z` = 6
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.
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 ______.
The shortest distance between the z-axis and the line x + y + 2z – 3 = 0 = 2x + 3y + 4z – 4, is ______.
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 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 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?
For skew lines, which expression gives \[SD\]?
