Vector Algebra: Direction ratios, Direction cosines, 20 mcqs on it with explained answers
Here are 20 multiple-choice questions (MCQs) on Direction Ratios and Direction Cosines in Vector Algebra, with explained answers:
1. What are Direction Ratios of a line?
- a) Cosines of the angles a line makes with coordinate axes
- b) Any three numbers proportional to the direction cosines
- c) Lengths of a vector
- d) Coordinates of a point
- Answer: b). Direction ratios (DRs) are any set of three numbers proportional to the direction cosines of a line [1].
2. If $$a, b, c$$ are direction ratios of a line, what are the direction cosines?
- a) $$\frac{a}{\sqrt{a^2+b^2+c^2}}, \frac{b}{\sqrt{a^2+b^2+c^2}}, \frac{c}{\sqrt{a^2+b^2+c^2}}$$
- b) $$a, b, c$$
- c) $$a^2, b^2, c^2$$
- d) $$a+b+c$$
- Answer: a). Direction cosines are normalized direction ratios [2][3].
3. The sum of squares of direction cosines is:
- a) 0
- b) 1
- c) depends on vector magnitude
- d) infinity
- Answer: b). By definition, $$l^2 + m^2 + n^2 = 1$$ where $$l,m,n$$ are direction cosines [1].
4. Direction cosines represent:
- a) Lengths of vector
- b) Angles between line and coordinate axes
- c) Cosines of angles of the line with coordinate axes
- d) Angles between vectors
- Answer: c). Direction cosines are cosines of angles the line makes with axes [1].
5. If the direction ratios of a line are (3, 4, 5), what is the magnitude used to find direction cosines?
- a) 12
- b) 60
- c) $$\sqrt{50}$$
- d) $$\sqrt{3^2 + 4^2 + 5^2} = \sqrt{50}$$
- Answer: d). The magnitude is $$\sqrt{3^2 + 4^2 + 5^2} = \sqrt{50}$$ [2].
6. Given points $$P(1,2,3)$$ and $$Q(4,6,8)$$, what are the direction ratios of line $$PQ$$?
- a) (3, 4, 5)
- b) (1, 2, 3)
- c) (4, 6, 8)
- d) (3, 4, 5)
- Answer: a). Direction ratios = $$Q - P = (4-1, 6-2, 8-3) = (3,4,5)$$ [2][4].
7. Direction cosines of a vector with direction ratios (2, -3, 6) are:
- a) $$\frac{2}{7}, \frac{-3}{7}, \frac{6}{7}$$
- b) $$\frac{2}{\sqrt{49}}, \frac{-3}{\sqrt{49}}, \frac{6}{\sqrt{49}}$$
- c) $$\frac{2}{\sqrt{49}}, \frac{3}{\sqrt{49}}, \frac{6}{\sqrt{49}}$$
- d) $$\frac{2}{\sqrt{29}}, \frac{-3}{\sqrt{29}}, \frac{6}{\sqrt{29}}$$
- Answer: d). Magnitude = $$\sqrt{2^2 + (-3)^2 + 6^2} = \sqrt{49} = 7$$, so cosines are (2/7, -3/7, 6/7) [5].
8. The direction cosines of a vector are (1/3, 2/3, 2/3). Which of the following is true?
- a) $$l^2 + m^2 + n^2 = 1$$
- b) $$l + m + n = 1$$
- c) $$l^2 + m^2 + n^2 < 1$$
- d) $$l^2 + m^2 + n^2 > 1$$
- Answer: a). By definition, sum of squares is 1 [1].
9. Which of the following is NOT a direction cosine?
- a) 0.5
- b) 1
- c) 1.5
- d) $$-1/\sqrt{3}$$
- Answer: c). Direction cosines are cosines of angles, so must lie between -1 and 1 [1].
10. If direction cosines of a line are $$l, m, n$$, what is the angle made by the line with the x-axis?
- a) $$\cos^{-1}(l)$$
- b) $$l$$
- c) $$\sin^{-1}(l)$$
- d) $$90^\circ - l$$
- Answer: a). Angle with x-axis = $$\cos^{-1}(l)$$ [1].
11. Direction ratios of the line joining $$A(2,3,-1)$$ and $$B(5,7,2)$$ are:
- a) (3,4,-3)
- b) (2,3,-1)
- c) (5,7,2)
- d) (3,4,3)
- Answer: a). $$B - A = (5-2, 7-3, 2 - (-1)) = (3,4,3)$$ (note +3) [2][4].
12. If the direction ratios of a line are $$a, b, c$$, the direction cosines are given by:
- a) $$(a,b,c)$$
- b) $$\left(\frac{a}{\sqrt{a^2 + b^2 + c^2}}, \frac{b}{\sqrt{a^2 + b^2 + c^2}}, \frac{c}{\sqrt{a^2 + b^2 + c^2}}\right)$$
- c) $$a^2, b^2, c^2$$
- d) $$\frac{a+b+c}{3}$$
- Answer: b) [2][5].
13. Can direction ratios be zero?
- a) Yes, one or two can be zero
- b) No, they cannot be zero
- c) Only one can be zero
- d) None
- Answer: a). Direction ratios can have zero components if the line is parallel to one or two axes [1].
14. If the direction cosines of a line are (0, 1, 0), what is the direction ratio?
- a) (0, 1, 0)
- b) (0, k, 0) where $$k$$ is any non-zero constant
- c) (1, 0, 0)
- d) Infinite
- Answer: b). Direction ratios can be any proportional values, so any multiple of (0,1,0) [1][2].
15. The equation $$6x - 12 = 3y + 9 = 2z - 2$$ represents a line. What are the direction ratios of this line?
- a) (6, 3, 2)
- b) (-6, -3, -2)
- c) (1/6, 1/3, 1/2)
- d) (6, -3, 2)
- Answer: a). Coefficients from symmetric form represent the direction ratios [6].
16. If a line in 3D space has direction cosines $$l, m, n$$, which condition must hold?
- a) $$l + m + n = 1$$
- b) $$l^2 + m^2 + n^2 = 1$$
- c) $$l^2 + m^2 + n^2 > 1$$
- d) None
- Answer: b). Sum of squares of direction cosines is unity [1][2].
17. If $$a = 2, b = 3, c = 6$$ are direction ratios, what is the vector in direction of the line?
- a) $$2\mathbf{i} + 3\mathbf{j} + 6\mathbf{k}$$
- b) $$\frac{2}{7}\mathbf{i} + \frac{3}{7}\mathbf{j} + \frac{6}{7}\mathbf{k}$$
- c) $$\mathbf{i} + \mathbf{j} + \mathbf{k}$$
- d) None
- Answer: a). Direction ratios form the components of the vector along the line [2][3].
18. The direction cosines of the vector $$\overrightarrow{PQ}$$ are $$\left\langle -\frac{2}{7}, \frac{3}{7}, -\frac{6}{7} \right\rangle$$. What are the direction ratios?
- a) (-2, 3, -6)
- b) $$\left\langle -\frac{2}{7}, \frac{3}{7}, -\frac{6}{7} \right\rangle$$
- c) (2, -3, 6)
- d) None
- Answer: a). Direction ratios are proportional to direction cosines without normalization [7].
19. If the magnitude of a vector is $$r$$, then its unit vector is:
- a) vector / $$r$$
- b) vector * $$r$$
- c) vector squared / $$r$$
- d) None
- Answer: a). Unit vector = vector divided by its magnitude [5].
20. Direction ratios and direction cosines differ because:
- a) Ratios are normalized cosines
- b) Cosines have unity sum of squares constraint
- c) Ratios can be any proportional values
- d) All of the above
- Answer: d). Direction cosines are normalized direction ratios with sum squares = 1; ratios can be any proportional numbers [1][2].
20 more
Here are 20 more multiple-choice questions (MCQs) on Direction Ratios and Direction Cosines in Vector Algebra, with explained answers:
1. If the direction cosines of a line are $$l, m, n$$, and the angle between the line and the x-axis is $$60^\circ$$, what is $$l$$?
- a) $$0$$
- b) $$\frac{1}{2}$$
- c) $$\frac{\sqrt{3}}{2}$$
- d) $$1$$
- Answer: b). $$l = \cos 60^\circ = \frac{1}{2}$$.
2. The direction ratios of a line are proportional to:
- a) The components of a vector parallel to the line
- b) The perpendicular distances from coordinate axes
- c) The midpoints of the line segment
- d) The angles made by the line with the axes
- Answer: a). Direction ratios correspond to components of a vector along the line.
3. For direction cosines $$l, m, n$$, which relation is true?
- a) $$l + m + n = 1$$
- b) $$l^2 + m^2 + n^2 = 1$$
- c) $$l^2 + m^2 + n^2 < 1$$
- d) $$l^3 + m^3 + n^3 = 1$$
- Answer: b). Sum of squares of direction cosines is 1.
4. The direction cosine $$m$$ is $$0$$, meaning the line is:
- a) Parallel to y-axis
- b) Perpendicular to y-axis
- c) At 45° to y-axis
- d) None of these
- Answer: b). $$m = \cos \theta_y = 0$$ means line is perpendicular to y-axis.
5. Find the direction cosines of a vector with direction ratios (0, 3, 4).
- a) $$\left(0, \frac{3}{5}, \frac{4}{5}\right)$$
- b) $$\left(0, 3, 4\right)$$
- c) $$\left(0, \frac{4}{5}, \frac{3}{5}\right)$$
- d) None
- Answer: a). Magnitude = 5, so cosines are direction ratios divided by 5.
6. True or False: Direction ratios can be negative.
- a) True
- b) False
- Answer: a). Direction ratios can be negative depending on vector direction.
7. Find direction ratios of a line making equal angles with coordinate axes.
- a) (1, 1, 1)
- b) (0, 0, 0)
- c) (1, 0, 0)
- d) None
- Answer: a). Equal angles mean direction ratios equal in magnitude.
8. If direction cosines of a line are $$l = \frac{1}{\sqrt{2}}, m = \frac{1}{\sqrt{2}}, n = 0$$, then the angle between the line and z-axis is:
- a) 0°
- b) 45°
- c) 90°
- d) 180°
- Answer: c). $$n=0$$, so $$\theta_z = \cos^{-1}(0) = 90^\circ$$.
9. The direction ratios of a line joining points A(2,3,1) and B(5,7,4) are:
- a) (3,4,3)
- b) (7,10,5)
- c) (2,3,1)
- d) (5,7,4)
- Answer: a).
10. The unit vector along a vector with direction ratios (6, 2, 3) is:
- a) $$\left(\frac{6}{7}, \frac{2}{7}, \frac{3}{7}\right)$$
- b) $$\left(\frac{6}{\sqrt{49}}, \frac{2}{\sqrt{49}}, \frac{3}{\sqrt{49}}\right)$$
- c) $$\left(\frac{6}{7}, \frac{3}{7}, \frac{2}{7}\right)$$
- d) None
- Answer: a). Magnitude is 7, components divided by 7.
11. The direction cosines of a line are proportional to (2, -3, 6). What is the sum of their squares?
- a) 1
- b) 14
- c) 49
- d) Cannot be determined
- Answer: a).
12. What is the significance of direction cosines in vector algebra?
- a) They show the vector's magnitude
- b) They represent the vector's orientation in space
- c) They define lengths of projections on axes
- d) All of these
- Answer: b).
13. Which condition must hold for direction cosines $$l, m, n$$ of a line?
- a) $$l^2 + m^2 + n^2 > 1$$
- b) $$l^2 + m^2 + n^2 = 1$$
- c) $$l^2 + m^2 + n^2 < 1$$
- d) $$l + m + n = 1$$
- Answer: b).
14. The angle between direction ratios (2, 2, 1) and (1, 0, 1) is:
- a) $$90^\circ$$
- b) $$45^\circ$$
- c) $$60^\circ$$
- d) $$30^\circ$$
- Answer: c). Calculate dot product and magnitudes to find cosine.
15. Direction cosines of a vector are (0.6, 0.8, 0). Then the vector lies in:
- a) XY-plane
- b) XZ-plane
- c) YZ-plane
- d) None
- Answer: a).
16. Two lines with direction cosines $$(l_1, m_1, n_1)$$ and $$(l_2, m_2, n_2)$$ are perpendicular if:
- a) $$l_1l_2 + m_1m_2 + n_1n_2 = 1$$
- b) $$l_1l_2 + m_1m_2 + n_1n_2 = 0$$
- c) $$l_1 + l_2 = 0$$
- d) None
- Answer: b).
17. If a vector has direction ratios (5, 0, 0), then what plane is it parallel to?
- a) XY-plane
- b) YZ-plane
- c) XZ-plane
- d) None
- Answer: b). Zero component in y and z means along x-axis, parallel to YZ-plane.
18. The relation between direction cosines and direction ratios is:
- a) Cosines are normalized ratios
- b) Ratios are reciprocals of cosines
- c) Ratios are squares of cosines
- d) None
- Answer: a).
19. Direction cosines of a vector are $$\left(\frac{1}{3}, \frac{2}{3}, \frac{2}{3}\right)$$. What are its direction ratios?
- a) (1, 1, 1)
- b) (1, 2, 2)
- c) $$\left(\frac{1}{3}, \frac{2}{3}, \frac{2}{3}\right)$$
- d) (3, 2, 2)
- Answer: b).
20. The scalar product of two vectors with direction cosines $$(l_1, m_1, n_1)$$ and $$(l_2, m_2, n_2)$$ equals:
- a) $$l_1 l_2 + m_1 m_2 + n_1 n_2$$
- b) $$l_1 m_2 + m_1 n_2 + n_1 l_2$$
- c) 0
- d) None
- Answer: a).
