Direction ratios and direction cosines - Question Bank

1. 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:
A) (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))
B) (1, b, c)
C) (a, b, c)
D) (1, 0, 0)
2. A line has direction ratios (1, 1, -2). What is the cosine of the angle it makes with the positive y-axis?
A) 1/sqrt(6)
B) -1/sqrt(6)
C) 1/sqrt(2)
D) -1/sqrt(2)
3. If a line makes an angle of 135 degrees with the positive x-axis, what is its direction cosine along the x-axis?
A) -1/sqrt(2)
B) 1/sqrt(2)
C) -1/2
D) 1/2
4. 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?
A) ab + bc + ca = 0
B) a+b+c = 0
C) a^2+b^2+c^2 = 0
D) a=b=c
5. What are the direction cosines of a line that passes through the origin and is equally inclined to the coordinate axes?
A) (1/sqrt(3), 1/sqrt(3), 1/sqrt(3))
B) (1, 0, 0)
C) (1/2, 1/2, 1/2)
D) (1/sqrt(2), 1/sqrt(2), 0)
6. 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?
A) 0
B) 1
C) l+m+n
D) cos(alpha)+cos(beta)+cos(gamma)
7. If a line has direction ratios (5, -12, 0), what is the cosine of the angle it makes with the positive x-axis?
A) 5/13
B) -12/13
C) 0
D) 5/sqrt(169)
8. What are the direction ratios of a line making angles 30, 60, and 90 degrees with the positive x, y, and z axes respectively?
A) (sqrt(3)/2, 1/2, 0)
B) (sqrt(3), 1, 0)
C) (1, 1, 0)
D) (30, 60, 90)
9. 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:
A) l1*l2 + m1*m2 + n1*n2
B) l1*l2 - m1*m2 - n1*n2
C) (l1+l2)/2 + (m1+m2)/2 + (n1+n2)/2
D) sqrt((l1-l2)^2 + (m1-m2)^2 + (n1-n2)^2)
10. A line has direction ratios (a, b, c). If b=0 and c=0, what is the direction of the line?
A) Parallel to the y-axis
B) Parallel to the z-axis
C) Parallel to the x-axis
D) It is not a valid line
11. What is the sum of the squares of the direction cosines of a line?
A) 0
B) 1
C) 2
D) Varies
12. If the direction ratios of a line are (k, 2k, 3k) for k != 0, what are its direction cosines?
A) (1/sqrt(14), 2/sqrt(14), 3/sqrt(14))
B) (k/sqrt(14k^2), 2k/sqrt(14k^2), 3k/sqrt(14k^2))
C) (1/sqrt(6), 2/sqrt(6), 3/sqrt(6))
D) (k, 2k, 3k)
13. Which of the following sets of numbers cannot be direction cosines of a line?
A) (1/sqrt(2), 0, 1/sqrt(2))
B) (1/3, 2/3, 2/3)
C) (0.6, 0.8, 0)
D) (1, 1, 1)
14. If a line has direction ratios (2, 3, -6), what is the cosine of the angle it makes with the positive z-axis?
A) -6/7
B) 6/7
C) 2/7
D) 3/7
15. What are the direction cosines of a line passing through the origin and the point (-3, 4, -12)?
A) (-3/13, 4/13, -12/13)
B) (3/13, -4/13, 12/13)
C) (-3/sqrt(253), 4/sqrt(253), -12/sqrt(253))
D) (3/sqrt(253), -4/sqrt(253), 12/sqrt(253))
16. 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?
A) 45 degrees
B) 60 degrees
C) 90 degrees
D) 120 degrees
17. The direction ratios of two lines are (1, 2, 3) and (2, 4, 6). What can be said about these lines?
A) They are perpendicular
B) They are parallel
C) They intersect at the origin
D) They are skew
18. If a line has direction ratios (3, 4, 0), what is its direction?
A) Parallel to the x-axis
B) Parallel to the y-axis
C) Parallel to the z-axis
D) Lies in the xy-plane perpendicular to the x-axis
19. What are the direction cosines of a line parallel to the vector i - j + k?
A) (1/sqrt(3), -1/sqrt(3), 1/sqrt(3))
B) (1, -1, 1)
C) (1/3, -1/3, 1/3)
D) (sqrt(3), -sqrt(3), sqrt(3))
20. 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?
A) k = 1/sqrt(a^2+b^2+c^2)
B) k = sqrt(a^2+b^2+c^2)
C) k = 1
D) k = a^2+b^2+c^2
21. If two lines have direction cosines (l1, m1, n1) and (l2, m2, n2) and are perpendicular, then:
A) l1*l2 + m1*m2 + n1*n2 = 0
B) l1*l2 + m1*m2 + n1*n2 = 1
C) l1*l2 = m1*m2 = n1*n2
D) l1/l2 = m1/m2 = n1/n2
22. A line has direction ratios (2, -1, 2). What is the cosine of the angle it makes with the positive y-axis?
A) -1/3
B) 2/3
C) -1/sqrt(5)
D) 2/sqrt(5)
23. What are the direction ratios of the z-axis?
A) (1, 0, 0)
B) (0, 1, 0)
C) (0, 0, 1)
D) (1, 1, 0)
24. 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:
A) 0
B) 1
C) 2
D) Depends on the angles
25. What is the relationship between the direction ratios (a,b,c) and the direction cosines (l,m,n) of a line?
A) l=a, m=b, n=c
B) 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)
C) l=sqrt(a^2+b^2+c^2)/a, m=sqrt(a^2+b^2+c^2)/b, n=sqrt(a^2+b^2+c^2)/c
D) l=1/a, m=1/b, n=1/c
26. If a line has direction cosines (2/3, -2/3, 1/3), what are its direction ratios?
A) (2, -2, 1)
B) (4/9, 4/9, 1/9)
C) (2/9, -2/9, 1/9)
D) (2/3, -2/3, 1/3)
27. What are the direction ratios of a line passing through points P(x1, y1, z1) and Q(x2, y2, z2)?
A) (x2-x1, y2-y1, z2-z1)
B) (x1-x2, y1-y2, z1-z2)
C) ((x2-x1)/PQ, (y2-y1)/PQ, (z2-z1)/PQ)
D) (x1+x2, y1+y2, z1+z2)
28. If l, m, n are direction cosines, which of the following is NOT possible?
A) l=1, m=0, n=0
B) l=1/2, m=1/2, n=1/sqrt(2)
C) l=1/3, m=2/3, n=2/3
D) l=1, m=1, n=1
29. A line has direction ratios (0, 5, 0). What is its direction?
A) Parallel to the x-axis
B) Parallel to the y-axis
C) Parallel to the z-axis
D) Along the line y=x in the xy-plane
30. 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) (-a, -b, -c)
B) (a, b, c)
C) (1/a, 1/b, 1/c)
D) (-1/a, -1/b, -1/c)
31. What are the direction cosines of a line that makes equal angles with all three coordinate axes?
A) (1/sqrt(3), 1/sqrt(3), 1/sqrt(3)) and its variations with negative signs
B) (1, 0, 0)
C) (1/2, 1/2, 1/2)
D) (1/sqrt(2), 1/sqrt(2), 0)
32. If the direction ratios of two lines are (1, 2, -2) and (x, 2x, 2x), and the lines are perpendicular, find x.
A) 1
B) -1
C) 0
D) 2
33. A line passes through the origin and has direction ratios (1, 1, 1). What is the angle it makes with the x-axis?
A) 30 degrees
B) 45 degrees
C) 60 degrees
D) 90 degrees
34. What is the sum of the products of the direction cosines of two perpendicular lines?
A) 0
B) 1
C) -1
D) Depends on the lines
35. If a line has direction ratios (1, 2, 3), what is the cosine of the angle it makes with the x-axis?
A) 1/sqrt(14)
B) 2/sqrt(14)
C) 3/sqrt(14)
D) 1/14
36. 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?
A) 1/2
B) -1/2
C) sqrt(3)/2
D) -sqrt(3)/2
37. If a line has direction ratios (0, 0, k) where k is non-zero, what is its direction?
A) Parallel to the x-axis
B) Parallel to the y-axis
C) Parallel to the z-axis
D) Along the origin
38. What are the direction cosines of a line perpendicular to both the x and y axes?
A) (0, 0, 1) or (0, 0, -1)
B) (1, 0, 0) or (-1, 0, 0)
C) (0, 1, 0) or (0, -1, 0)
D) (1, 1, 0)
39. Two lines have direction ratios (a1, b1, c1) and (a2, b2, c2). Under what condition are they perpendicular?
A) a1*a2 + b1*b2 + c1*c2 = 0
B) a1*a2 + b1*b2 + c1*c2 = 1
C) a1*a2 = b1*b2 = c1*c2
D) a1/a2 = b1/b2 = c1/c2
40. If a line has direction ratios proportional to 1, -1, 2, what are its direction cosines?
A) (1/sqrt(6), -1/sqrt(6), 2/sqrt(6))
B) (1/2, -1/2, 2/2)
C) (1/3, -1/3, 2/3)
D) (1/sqrt(2), -1/sqrt(2), 2/sqrt(2))
41. What are the direction cosines of a line parallel to the vector 2i - 3j + 6k?
A) (2/7, -3/7, 6/7)
B) (2/sqrt(7), -3/sqrt(7), 6/sqrt(7))
C) (2/49, -3/49, 6/49)
D) (7/2, -7/3, 7/6)
42. If the direction ratios of a line are 2, 3, 6, what is the magnitude of its direction cosine vector?
A) 1
B) 7
C) sqrt(7)
D) 49
43. A line has direction ratios -18, 12, -4. What are its direction cosines?
A) (-9/11, 6/11, -2/11)
B) (9/11, -6/11, 2/11)
C) (-18/11, 12/11, -4/11)
D) (18/11, -12/11, 4/11)
44. If a line makes angles alpha, beta, gamma with the positive x, y, and z axes respectively, then what are its direction cosines?
A) (cos(alpha), cos(beta), cos(gamma))
B) (sin(alpha), sin(beta), sin(gamma))
C) (tan(alpha), tan(beta), tan(gamma))
D) (alpha, beta, gamma)
45. What are the direction cosines of the y-axis?
A) (1, 0, 0)
B) (0, 1, 0)
C) (0, 0, 1)
D) (0, -1, 0)
46. What are the direction ratios of the x-axis?
A) (1, 0, 0)
B) (0, 1, 0)
C) (0, 0, 1)
D) (1, 1, 1)
47. If a, b, c are direction ratios of a line, what are its direction cosines?
A) (a/sqrt(a^2+b^2+c^2), b/sqrt(a^2+b^2+c^2), c/sqrt(a^2+b^2+c^2))
B) (a^2, b^2, c^2)
C) (sqrt(a^2+b^2+c^2)/a, sqrt(a^2+b^2+c^2)/b, sqrt(a^2+b^2+c^2)/c)
D) (a, b, c)
48. What are the direction cosines of a line passing through the origin and the point (x, y, z)?
A) (x/sqrt(x^2+y^2+z^2), y/sqrt(x^2+y^2+z^2), z/sqrt(x^2+y^2+z^2))
B) (x, y, z)
C) (x^2, y^2, z^2)
D) (sqrt(x^2+y^2+z^2)/x, sqrt(x^2+y^2+z^2)/y, sqrt(x^2+y^2+z^2)/z)
49. If l, m, n are the direction cosines of a line, what is the relationship between them?
A) l + m + n = 1
B) l^2 + m^2 + n^2 = 0
C) l^2 + m^2 + n^2 = 1
D) l^2 + m^2 + n^2 = -1
50. What are direction ratios of a line?
A) The squares of the direction cosines
B) The direction cosines themselves
C) Any three numbers proportional to the direction cosines
D) The sum of the squares of the direction cosines
All Question Banks
E-Book Direction ratios and direction cosines One Line Questions Direction ratios and direction cosines Online Test Direction ratios and direction cosines