Advertisements
Advertisements
Question
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.
Advertisements
Solution
a. Given, lines are: `vecr = λ(hati + 2hatj - hatk)` and `vecr = (3hati + 3hatj) + μ(2hati + hatj + hatk)`
We know that, shortest distance between the lines
`vecr_1 = veca + λb_1` and `vecr = veca_2 + λvecb_1` is d = `(|(veca_2 - veca_1).(vecb_1 xx vecb_2)|)/(|vecb_1 xx vecb_2|)`
Here, `veca_1 = 0, veca_2 = (3hati + 3hatj)`
`vecb_1 = hati + 2hatj - hatk`
and `vecb_2 = 2hati + hatj + hatk`
∴ `veca_2 - veca_1 = (3hati + 3hatj) - 0 = 3hati + 3hatj`
`vecb_1 xx vecb_2 = |(hati, hatj, hatk),(1, 2, -1),(2, 1, 1)|`
= `hati(2 + 1) - hatj(1 + 2) + hatk(1 - 4)`
= `3hati - 3hatj - 3hatk`
and `|vecb_1 xx vecb_2| = sqrt(3^2 + (-3)^2 + (-3)^2)`
= `sqrt(9 + 9 + 9)`
= `3sqrt(3)`
Also, `(veca_2 - veca_1).(vecb_1 xx vecb_2) = (3hati + 3hatj).(3hati - 3hatj - 3hatk)`
= 9 – 9
= 0
d = `0/(3sqrt(3))` = 0
Thus, distance between lines is 0.
b. We have, `vecr = λ(hati + 2hatj - hatk)` ...(i)
and `vecr = 3hati + 3hatj + μ(2hati + hatj + hatk)`
or `vecr = (3 + 2μ)hati + (3 + μ)hatj + μhatk` ...(ii)
Now, from equation (i) and equation (ii), we get
`λ(hati + 2hatj - hatk) = (3 + 2μ)hati + (3 + μ)hatj + μhatk`
On comparing both sides, we get
3 + 2µ = λ, 3 + µ = 2λ and µ = –λ
On solving for values of λ and µ, we get
λ = 1 and µ = –1
From equation (i), we get `vecr = hati + 2hatj - hatk`
`xhati + yhatj + zhatk = hati + 2hatj - hatk`
So, required point is (1, 2, –1).
APPEARS IN
RELATED QUESTIONS
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
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 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 `vecr = (4hati - hatj) + lambda(hati+2hatj-3hatk)` and `vecr = (hati - hatj + 2hatk) + mu(2hati + 4hatj - 5hatk)`
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. |
The fuel cost for 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)`
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 \[SD\]?
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 \[\vec a_2\]?
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 \[\vec r=(3\hat i+3\hat j-5\hat k)+\mu(2\hat i+3\hat j+6\hat k)\]?
