Shortest distance between skew lines - One Line Questions
1.
What is the determinant form to calculate the shortest distance between two skew lines given by their symmetric equations? —
| x2-x1 y2-y1 z2-z1 | / sqrt( (b1c2-b2c1)^2 + (c1a2-c2a1)^2 + (a1b2-a2b1)^2 )
2.
If two lines are given in Cartesian form as (x-x1)/a1 = (y-y1)/b1 = (z-z1)/c1 and (x-x2)/a2 = (y-y2)/b2 = (z-z2)/c2, what is the formula for the shortest distance? —
|( (x2-x1, y2-y1, z2-z1) . (a1, b1, c1) x (a2, b2, c2) )| / |(a1, b1, c1) x (a2, b2, c2)|
3.
If two lines are given by vector equations r = a1 + λb1 and r = a2 + μb2, what is the formula for the shortest distance between them? —
|(a1 - a2) . (b1 x b2)| / |b1 x b2|
4.
If the lines r = a1 + λb1 and r = a2 + μb2 are perpendicular, the shortest distance is: —
|(a2 - a1) . (b1 x b2)| / |b1 x b2|
5.
What is the formula for the shortest distance between two parallel lines r = a1 + λb and r = a2 + μb? —
|(a2 - a1) x b| / |b|
6.
The shortest distance between the origin and the line r = (i + j + k) + λ(2i + 3j + 4k) is: —
√14 / √29
7.
What is the shortest distance between the lines x = 1, y = 1, z = 1+t and x = 2, y = 2, z = 2+s? —
√2
8.
The shortest distance between the lines x = 3, y = 2, z = 1 + t and x = 1, y = 1, z = 2 + s is: —
√2
9.
Calculate the magnitude of (b1 x b2) for the lines L1: r = (i + 2j + k) + λ(2i - j + k) and L2: r = (2i + j - k) + μ(i + j - k). —
√22
10.
What is the shortest distance between the lines x = 1, y = 2, z = 3 + t and x = 4, y = 5, z = 6 + s? —
√27
11.
What is the shortest distance between the y-axis and the line x=1, y=t, z=2? —
√5
12.
The shortest distance between the lines r = (3i + 2j - 4k) + λ(i + 2j + 2k) and r = (5i - 2j + k) + μ(3i + 2j + k) is: —
3/√29
13.
The shortest distance between the lines x/1 = y/2 = z/3 and x = 3t, y = 4t, z = 5t is: —
0
14.
If two lines are given by x = 2y = 3z and x+1 = y-2 = z+3, what is the shortest distance between them? —
0
15.
The shortest distance between the lines x = 1 + t, y = 2 + t, z = 3 + t and x = 2 + s, y = 3 + s, z = 4 + s is: —
0
16.
What is the shortest distance between the lines x = t, y = 0, z = 0 and x = 0, y = s, z = 0? —
0
17.
The shortest distance between the lines x = 1 + 2t, y = 2 + 3t, z = 3 + 4t and x = 2 + 3s, y = 4 + 4s, z = 5 + 5s is: —
0
18.
If the shortest distance between two skew lines is given by d = |(a2 - a1) . (b1 x b2)| / |b1 x b2|, and (a2 - a1) is perpendicular to (b1 x b2), what is the shortest distance? —
0
19.
The shortest distance between the lines x = 1, y = 2, z = 3 and x = 1, y = 2, z = 3 is: —
0
20.
The vector form of a line is r = a + λb. If a line passes through the origin, what is the value of vector a? —
0 (zero vector)
21.
Find the shortest distance between the lines (x-1)/2 = (y-2)/3 = (z-3)/4 and (x-2)/3 = (y-4)/4 = (z-5)/5. —
1/√2
22.
For the lines L1: r = (i + 2j + k) + λ(2i - j + k) and L2: r = (2i + j - k) + μ(i + j - k), what is the vector (b1 x b2)? —
2i + 3j + 3k
23.
What is the shortest distance between the lines x=1+2t, y=2-t, z=t and x=2+s, y=1+s, z=-s? —
5/√14
24.
The shortest distance between the lines r = (2i - j + k) + λ(i - 2j + k) and r = (i - j + 2k) + μ(2i + j - k) is: —
3/√14
25.
Calculate the shortest distance between the lines L1: r = (i + 2j + k) + λ(2i - j + k) and L2: r = (2i + j - k) + μ(i + j - k). —
4/√22
26.
Consider the lines L1: r = i + 2j + k + λ(2i - j + k) and L2: r = 2i + j - k + μ(i + j - k). What is the shortest distance between the points on these lines that are closest to each other? —
4/√22
27.
In the shortest distance formula for skew lines, what does b1 x b2 represent? —
A vector perpendicular to both b1 and b2
28.
If two lines are parallel, with direction vector b, and points a1 and a2, the shortest distance is the length of the projection of (a2 - a1) onto: —
b
29.
What is the condition for two lines r = a1 + λb1 and r = a2 + μb2 to be skew? —
b1 and b2 are not parallel and a2 - a1 is not parallel to b1 or b2
30.
Let the equations of two skew lines be r = a1 + λb1 and r = a2 + μb2. The shortest distance is the length of the projection of the vector (a2 - a1) onto which vector? —
b1 x b2
31.
If b1 and b2 are the direction vectors of two skew lines, the direction vector of the common perpendicular is proportional to: —
b1 x b2
32.
What is the condition for the shortest distance between two lines r = a1 + λb1 and r = a2 + μb2 to be zero? —
(a2 - a1) . (b1 x b2) = 0
33.
If the shortest distance between two skew lines is d, and the angle between the vector (a2 - a1) and (b1 x b2) is α, then: —
d = |(a2 - a1) . (b1 x b2)| / |b1 x b2|
34.
Consider the lines L1: r = (i + 2j + k) + λ(2i - j + k) and L2: r = (2i + j - k) + μ(i + j - k). What is the vector (a2 - a1)? —
i - j - 2k
35.
What is the magnitude of the vector cross product of two non-parallel vectors? —
It is always positive
36.
If the shortest distance between two skew lines is given by d, and the angle between their direction vectors is θ, what is the relationship between d and cos(θ)? —
No direct relationship
37.
What type of lines are necessary to calculate the shortest distance between them? —
Skew lines
38.
The shortest distance between two skew lines is always: —
Non-negative
39.
Which of the following pairs of lines are skew? —
r = i + λj and r = 2i + μ(i+j)
40.
Consider the lines x = 1, y = 2, z = 3 + t and x = 4, y = 5, z = 6 + s. These lines are: —
Skew
41.
If the lines are given by r = a1 + λb1 and r = a2 + μb2, and b1 is parallel to b2, then the lines are: —
Parallel
42.
If the direction ratios of two lines are proportional, what is the shortest distance between them if they are not coincident? —
Infinite
43.
What is the geometric interpretation of the shortest distance between two skew lines? —
The length of the common perpendicular segment
44.
If a1, b1, c1 and a2, b2, c2 are the direction ratios of two lines, and x1, y1, z1 and x2, y2, z2 are points on them, the shortest distance is zero if: —
The lines are intersecting
45.
If the shortest distance between two skew lines is zero, what does it imply? —
The lines intersect
46.
What does it mean if the shortest distance between two lines is calculated using the formula for parallel lines, but the direction vectors are not parallel? —
The calculation is invalid
47.
For skew lines r = a1 + λb1 and r = a2 + μb2, what does the term (a2 - a1) represent in the shortest distance formula? —
The vector connecting a point on the first line to a point on the second line
48.
Which of the following is NOT a method to find the shortest distance between skew lines? —
Finding the intersection point and measuring distance
49.
Two lines are given by r = (i + j) + λ(2i - j + k) and r = (2i + j - k) + μ(i + j - k). Are these lines skew? —
Yes, because their direction vectors are not parallel