Advertisements
Advertisements
प्रश्न
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 ______.
विकल्प
coincident
skew
intersecting
parallel
उत्तर
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 parallel.
Explanation:
Vectors `2hati + 3hatj - 6hatk` and `6hati + 9hatj - 18hatk` are parallel and the fixed point `hati + hatj - hatk` on the line `vecr = hati + hatj - hatk + λ(2hati + 3hatj - 6hatk)` does not satisfy the other line `vecr = 2hati - hatj - hatk + μ(6hati + 9hatj - 18hatk)`; where λ and μ are scalars.
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:
`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 = (hati + 2hatj + 3hatk) + lambda(hati - 3hatj + 2hatk)` and `vecr = 4hati + 5hatj + 6hatk + mu(2hati + 3hatj + 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 `(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. |
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`
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`
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)`
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 ______.
Find the distance between the lines:
`vecr = (hati + 2hatj - 4hatk) + λ(2hati + 3hatj + 6hatk)`;
`vecr = (3hati + 3hatj - 5hatk) + μ(4hati + 6hatj + 12hatk)`
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.
