Direction ratios and direction cosines - One Line Questions
1.
A line has direction ratios (2, -1, 2). What is the cosine of the angle it makes with the positive y-axis? —
-1/3
2.
If a line makes an angle of 135 degrees with the positive x-axis, what is its direction cosine along the x-axis? —
-1/sqrt(2)
3.
If a line has direction ratios (2, 3, -6), what is the cosine of the angle it makes with the positive z-axis? —
-6/7
4.
What are the direction cosines of a line passing through the origin and the point (-3, 4, -12)? —
5.
A line has direction ratios -18, 12, -4. What are its direction cosines? —
(-9/11, 6/11, -2/11)
6.
If a line has direction ratios (a, b, c), what are the direction ratios of the line parallel to it in the opposite direction? —
(-a, -b, -c)
7.
What are the direction cosines of a line perpendicular to both the x and y axes? —
(0, 0, 1) or (0, 0, -1)
8.
If a line has direction ratios (a, b, c), then the direction ratios of the line bisecting the angle between the positive x-axis and the line are proportional to: —
(1 + a/sqrt(a^2+b^2+c^2), b/sqrt(a^2+b^2+c^2), c/sqrt(a^2+b^2+c^2))
9.
What are the direction ratios of the x-axis? —
(1, 0, 0)
10.
What are the direction cosines of the y-axis? —
(0, 1, 0)
11.
What are the direction ratios of the z-axis? —
(0, 0, 1)
12.
If the direction ratios of a line are (k, 2k, 3k) for k != 0, what are its direction cosines? —
(k/sqrt(14k^2), 2k/sqrt(14k^2), 3k/sqrt(14k^2))
13.
Which of the following sets of numbers cannot be direction cosines of a line? —
(1, 1, 1)
14.
What are the direction cosines of a line parallel to the vector i - j + k? —
(1/sqrt(3), -1/sqrt(3), 1/sqrt(3))
15.
What are the direction cosines of a line that passes through the origin and is equally inclined to the coordinate axes? —
(1/sqrt(3), 1/sqrt(3), 1/sqrt(3))
16.
What are the direction cosines of a line that makes equal angles with all three coordinate axes? —
(1/sqrt(3), 1/sqrt(3), 1/sqrt(3)) and its variations with negative signs
17.
If a line has direction ratios proportional to 1, -1, 2, what are its direction cosines? —
(1/sqrt(6), -1/sqrt(6), 2/sqrt(6))
18.
If a line has direction cosines (2/3, -2/3, 1/3), what are its direction ratios? —
(2, -2, 1)
19.
What are the direction cosines of a line parallel to the vector 2i - 3j + 6k? —
(2/7, -3/7, 6/7)
20.
If a, b, c are direction ratios of a line, what are its direction cosines? —
(a/sqrt(a^2+b^2+c^2), b/sqrt(a^2+b^2+c^2), c/sqrt(a^2+b^2+c^2))
21.
If a line makes angles alpha, beta, gamma with the positive x, y, and z axes respectively, then what are its direction cosines? —
(cos(alpha), cos(beta), cos(gamma))
22.
What are the direction ratios of a line making angles 30, 60, and 90 degrees with the positive x, y, and z axes respectively? —
(sqrt(3), 1, 0)
23.
What are the direction cosines of a line passing through the origin and the point (x, y, z)? —
(x/sqrt(x^2+y^2+z^2), y/sqrt(x^2+y^2+z^2), z/sqrt(x^2+y^2+z^2))
24.
What are the direction ratios of a line passing through points P(x1, y1, z1) and Q(x2, y2, z2)? —
(x2-x1, y2-y1, z2-z1)
25.
What is the sum of the squares of the direction cosines of a line? —
1
26.
If l, m, n are direction cosines, what is the value of l*cos(alpha) + m*cos(beta) + n*cos(gamma) where alpha, beta, gamma are angles with axes? —
1
27.
What is the sum of the products of the direction cosines of two perpendicular lines? —
0
28.
If a line makes angles alpha, beta, gamma with the positive x, y, z axes, then cos^2(alpha) + cos^2(beta) + cos^2(gamma) is equal to: —
1
29.
If the direction ratios of a line are 2, 3, 6, what is the magnitude of its direction cosine vector? —
1
30.
If the direction ratios of two lines are (1, 2, -2) and (x, 2x, 2x), and the lines are perpendicular, find x. —
-1
31.
The direction cosines of a line are (l, m, n). If the line makes an angle of 60 degrees with the positive x-axis, what is the value of l? —
1/2
32.
If a line has direction ratios (1, 2, 3), what is the cosine of the angle it makes with the x-axis? —
1/sqrt(14)
33.
A line has direction ratios (1, 1, -2). What is the cosine of the angle it makes with the positive y-axis? —
1/sqrt(6)
34.
A line passes through the origin and has direction ratios (1, 1, 1). What is the angle it makes with the x-axis? —
60 degrees
35.
If a line makes angles of 45 degrees and 60 degrees with the positive x and y axes respectively, what is the angle it makes with the positive z-axis? —
120 degrees
36.
If a line has direction ratios (5, -12, 0), what is the cosine of the angle it makes with the positive x-axis? —
5/13
37.
Two lines have direction ratios (a1, b1, c1) and (a2, b2, c2). Under what condition are they perpendicular? —
a1*a2 + b1*b2 + c1*c2 = 0
38.
If a line has direction ratios (a, b, c), and a line has direction ratios (b, c, a), what is the condition for them to be perpendicular? —
ab + bc + ca = 0
39.
If a line has direction ratios (a, b, c), what is the value of the constant k such that (ka, kb, kc) are its direction cosines? —
k = 1/sqrt(a^2+b^2+c^2)
40.
If l, m, n are the direction cosines of a line, what is the relationship between them? —
l^2 + m^2 + n^2 = 1
41.
If l, m, n are direction cosines, which of the following is NOT possible? —
l=1, m=1, n=1
42.
What is the relationship between the direction ratios (a,b,c) and the direction cosines (l,m,n) of a line? —
l=a/sqrt(a^2+b^2+c^2), m=b/sqrt(a^2+b^2+c^2), n=c/sqrt(a^2+b^2+c^2)
43.
If the direction cosines of two lines are (l1, m1, n1) and (l2, m2, n2) and the angle between them is theta, then cos(theta) is given by: —
l1*l2 + m1*m2 + n1*n2
44.
If two lines have direction cosines (l1, m1, n1) and (l2, m2, n2) and are perpendicular, then: —
l1*l2 + m1*m2 + n1*n2 = 0
45.
If a line has direction ratios (0, 0, k) where k is non-zero, what is its direction? —
Parallel to the z-axis
46.
A line has direction ratios (0, 5, 0). What is its direction? —
Parallel to the y-axis
47.
If a line has direction ratios (3, 4, 0), what is its direction? —
Lies in the xy-plane perpendicular to the x-axis
48.
A line has direction ratios (a, b, c). If b=0 and c=0, what is the direction of the line? —
Parallel to the x-axis
49.
What are direction ratios of a line? —
Any three numbers proportional to the direction cosines
50.
The direction ratios of two lines are (1, 2, 3) and (2, 4, 6). What can be said about these lines? —
They are parallel