Equation of a line in space and skew lines - One Line Questions
1.
Find the point where the line (x-2)/3 = (y+1)/2 = (z-3)/1 intersects the xy-plane. —
(5, 1, 5)
2.
Find the foot of the perpendicular from the point (1, 2, 3) to the line (x-2)/3 = (y+1)/2 = (z-3)/1. —
(5, 1, 5)
3.
For the lines to be coplanar, the scalar triple product of the vector joining any point on the first line to any point on the second line and the direction vectors of the two lines must be zero. What is this condition? —
(a2 - a1) . (b1 x b2) = 0
4.
What is the condition for the lines x = a1 + λb1 and x = a2 + μb2 to be coplanar? —
(a2 - a1) . (b1 x b2) = 0
5.
Find the equation of the line passing through (1, 2, 3) and having direction ratios 4, 5, 6. —
(x - 1)/4 = (y - 2)/5 = (z - 3)/6
6.
What is the equation of the line passing through (1, 1, 1) and parallel to the line 2x - 1 = 3y - 1 = 4z - 1? —
(x - 1)/6 = (y - 1)/4 = (z - 1)/3
7.
The equation of a line passing through (2, 3, 4) and parallel to vector 3i + 2j - 8k in Cartesian form is: —
(x - 2)/3 = (y - 3)/2 = (z - 4)/-8
8.
What is the Cartesian form of the equation of a line passing through two points (x1, y1, z1) and (x2, y2, z2)? —
(x - x1)/(x2 - x1) = (y - y1)/(y2 - y1) = (z - z1)/(z2 - z1)
9.
What is the equation of the line passing through (a, b, c) and parallel to the x-axis? —
(x-a)/1 = (y-b)/0 = (z-c)/0
10.
What is the shortest distance between two skew lines r = a1 + λb1 and r = a2 + μb2? —
|(a2 - a1) . (b1 x b2)| / |b1 x b2|
11.
What is the shortest distance between two parallel lines r = a1 + λb and r = a2 + μb? —
|(a2 - a1) x b| / |b|
12.
The lines (x - x1)/l1 = (y - y1)/m1 = (z - z1)/n1 and (x - x2)/l2 = (y - y2)/m2 = (z - z2)/n2 intersect if: —
|(x2-x1)(m1n2 - m2n1) + (y2-y1)(n1l2 - n2l1) + (z2-z1)(l1m2 - l2m1)| = 0
13.
Find the shortest distance between the lines r = i + 2j + 3k + λ(2i + 3j + 4k) and r = 2i + 3j + 4k + μ(3i + 4j + 5k). —
0
14.
Find the angle between the line joining (2, 3, 4) to (3, 4, 5) and the line joining (1, 2, 3) to (2, 3, 4). —
0 degrees
15.
The direction ratios of a line are proportional to (1, -2, 3). What are its direction cosines? —
1/√14, -2/√14, 3/√14
16.
Find the value of λ for which the lines (x-1)/2 = (y-2)/3 = (z-3)/4 and (x-2)/λ = (y-4)/6 = (z-5)/8 are parallel. —
2
17.
Find the direction ratios of the line passing through points P(2, 4, 6) and Q(3, 5, 7). —
1, 1, 1
18.
If the lines (x-1)/a = (y-2)/b = (z-3)/c and (x-2)/2 = (y-3)/3 = (z-4)/4 are perpendicular, then: —
2a + 3b + 4c = 0
19.
If the lines (x-1)/l = (y-2)/m = (z-3)/n and (x-2)/2 = (y-3)/3 = (z-4)/4 are coplanar, then: —
|(2-1)(3n-4m) + (3-2)(4l-2n) + (4-3)(2m-3l)| = 0
20.
Find the projection of the line segment joining (1, 2, 3) and (4, 5, 6) onto the line with direction ratios (2, 1, -2). —
9
21.
The lines x = x1 + at, y = y1 + bt, z = z1 + ct and x = x2 + du, y = y2 + eu, z = z2 + fu are parallel if: —
a/d = b/e = c/f
22.
If two lines are given by r = a1 + λb1 and r = a2 + μb2, what is the condition for them to be parallel? —
b1 x b2 = 0
23.
Two lines are parallel if their direction vectors are proportional. What is the condition for direction vectors b1 and b2? —
b1 = λb2 for some scalar λ
24.
Two lines are perpendicular if the dot product of their direction vectors is zero. If direction vectors are b1 and b2, what is the condition? —
b1 . b2 = 0
25.
If two lines are given by r = a1 + λb1 and r = a2 + μb2, what is the condition for them to be perpendicular? —
b1 . b2 = 0
26.
Find the angle between the line (x-2)/3 = (y+1)/2 = (z-3)/1 and the line (x+1)/2 = (y-2)/1 = (z+2)/3. —
cos⁻¹(10 / sqrt(14)sqrt(14))
27.
Find the angle between the lines with direction ratios (1, 2, 3) and (2, -1, 1). —
cos⁻¹(1 / sqrt(14)sqrt(6))
28.
If the direction cosines of a line are l, m, n, then which of the following is true? —
l² + m² + n² = 1
29.
What is the condition for the lines (x - x1)/l1 = (y - y1)/m1 = (z - z1)/n1 and (x - x2)/l2 = (y - y2)/m2 = (z - z2)/n2 to be skew? —
|(x2-x1)(m1n2 - m2n1) + (y2-y1)(n1l2 - n2l1) + (z2-z1)(l1m2 - l2m1)| != 0
30.
Find the relation between the direction cosines l, m, n of two perpendicular lines. —
l1l2 + m1m2 + n1n2 = 0
31.
What are skew lines? —
Lines that are not parallel and do not intersect.
32.
The lines (x - 1)/(-1) = (y - 2)/2 = (z - 3)/3 and (x - 1)/3 = (y - 2)/2 = (z - 3)/(-1) are: —
Perpendicular
33.
The lines r = (i + j) + λ(i - j + k) and r = (2i + k) + μ(i - 2j + k) are: —
Skew
34.
The lines x=t, y=2t, z=3t and x=u, y=3u, z=4u are: —
Skew
35.
The lines (x-1)/2 = (y-2)/3 = (z-3)/4 and (x-1)/3 = (y-2)/4 = (z-3)/5 are: —
Skew
36.
What is the vector form of the equation of a line passing through a point with position vector 'a' and parallel to a vector 'b'? —
r = a + λb
37.
The equation of a line passing through (2, 3, 4) and parallel to vector 3i + 2j - 8k in vector form is: —
r = (2i + 3j + 4k) + λ(3i + 2j - 8k)
38.
The vector equation of the line passing through (1, 2, 3) and parallel to the vector 2i + 3j + 4k is: —
r = (i + 2j + 3k) + λ(2i + 3j + 4k)
39.
What is the vector form of the equation of a line passing through two points with position vectors 'a' and 'b'? —
r = (1 - λ)a + λb
40.
What is the vector equation of the line passing through the origin and the point (2, 3, 4)? —
r = λ(2i + 3j + 4k)
41.
The equation of the line passing through the origin and perpendicular to the lines with direction ratios (1, 2, 3) and (2, -1, 1) is: —
r = λ(5i + 5j + k)
42.
Find the shortest distance between the origin and the line (x-2)/3 = (y+1)/2 = (z-3)/1. —
sqrt(38 / 14)
43.
Find the shortest distance between the lines r = i + j + k + λ(2i - j + k) and r = 2i - j + k + μ(i + 2j - k). —
sqrt(2) / 3
44.
Find the shortest distance between the lines x = 1 + 2t, y = 2 + 3t, z = 3 + 4t and x = 2 + 3u, y = 4 + 4u, z = 5 + 5u. —
sqrt(2) / sqrt(14)
45.
Two lines are given by r = a1 + λb1 and r = a2 + μb2. If a1 = i + 2j - k, b1 = 2i + j - k, a2 = 2i - j + k, b2 = i + 2j + k. Find the shortest distance. —
sqrt(21) / 7
46.
What is the distance of the point (1, 2, 3) from the line (x-2)/3 = (y+1)/2 = (z-3)/1? —
sqrt(38 / 14)
47.
If the shortest distance between two skew lines is zero, what does it imply? —
The lines are coplanar (intersecting).
48.
What is the condition for two lines in space to be coplanar? —
The shortest distance between them is zero.
49.
Find the equation of the plane containing the line (x-1)/2 = (y-2)/3 = (z-3)/4 and parallel to the line x/3 = y/4 = z/5. —
x - 2y + z = 0
50.
What is the Cartesian form of the equation of a line passing through a point (x1, y1, z1) and direction ratios (l, m, n)? —
(x - x1)/l = (y - y1)/m = (z - z1)/n