Advertisements
Advertisements
Question
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)`
Advertisements
Solution
We know that the lines
`(x−x_1)/l_1=(y−y_1)/m_1=(z−z_1)/n_1 and (x−x_2)/l_2=(y−y_2)/m_2=(z−z_2)/n_2` are coplanar, if
`|[x_2-x_1,y_2-y_1,z_2-z_1],[1_1,m_1,n_1],[l_2,m_2,n_2]|=0`
Now, the equations of the given lines are
`(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)`
These can be rewritten as
`therefore |[(b-c)-(a-d),b-a,(b+c)-(a+d)],[alpha-beta,alpha,alpha+delta],[beta-gama,beta,beta+gama]|`
`=|[2(b-a),(b-a),(b+c)-(a+d)],[2alpha,alpha,alpha+delta],[2beta,beta,beta+gama]| [`
`=2|[(b-a),b-a,(b+c)-(a+d)],[alpha,alpha,alpha+delta],[beta,beta,beta+gama]| `
=0 [C1 and C2 are identical]
Hence, the given lines are coplanar.
APPEARS IN
RELATED QUESTIONS
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
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 = (1-t)hati + (t - 2)hatj + (3 -2t)hatk` and `vecr = (s+1)hati + (2s + 1)hatk`.
Find the shortest distance between lines `vecr = 6hati + 2hatj + 2hatk + lambda(hati - 2hatj + 2hatk)` and `vecr =-4hati - hatk + mu(3hati - 2hatj - 2hatk)`.
|
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)`
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)`
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
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 ______.
Find the distance between the lines:
`vecr = (hati + 2hatj - 4hatk) + λ(2hati + 3hatj + 6hatk)`;
`vecr = (3hati + 3hatj - 5hatk) + μ(4hati + 6hatj + 12hatk)`
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 ______.
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.
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.
What does the concept of shortest distance help in finding when two lines do not meet directly?
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 \[\vec b_1\times\vec b_2\]?
What is the distance between \[\vec r=(\hat i+\hat j)+\lambda(2\hat i-\hat j+\hat k)\] and \[\vec r=(2\hat i+\hat j-\hat k)+\mu(3\hat i-5\hat j+2\hat k)\]?
What is \[SD\] for intersecting lines?
For skew lines, which expression gives \[SD\]?
