Three Dimensional Geometry

Comprehensive Question Bank & Study Notes

Multiple Choice Questions

1 Competency APPLY MCQ_SINGLE 1 Mark(s) 2025 AISSCE(Board Exam)
#673
The equation of a line parallel to the vector \(3\hat{i}+\hat{j}+2\hat{k}\) and passing through the point \((4, -3, 7)\) is:
(A) \(x=4t+3, y=-3t+1, z=7t+2\)
(B) \(x=3t+4, y=t+3, z=2t+7\)
(C) \(x=3t+4, y=t-3, z=2t+7\)
(D) \(x=3t+4, y=-t+3, z=2t+7\)
MEDIUM
2 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2025 AISSCE(Board Exam)
#672
The line \(x=1+5\mu\), \(y=-5+\mu\), \(z=-6-3\mu\) passes through which of the following point ?
(A) \((1, -5, 6)\)
(B) \((1, 5, 6)\)
(C) \((1, -5, -6)\)
(D) \((-1, -5, 6)\)
MEDIUM
3 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2025 AISSCE(Board Exam)
#671
If P is a point on the line segment joining (3, 6, -1) and (6, 2, -2) and y-coordinate of P is 4, then its z-coordinate is:
(A) \(-\frac{3}{2}\)
(B) 0
(C) 1
(D) \(\frac{3}{2}\)
MEDIUM
4 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2024 AISSCE(Board Exam)
#670
If the direction cosines of a line are \(\sqrt{3}k, \sqrt{3}k\), \(\sqrt{3}k,\) then the value of k is:
(A) \(\pm1\)
(B) \(\pm\sqrt{3}\)
(C) \(\pm3\)
(D) \(\pm\frac{1}{3}\)
MEDIUM
5 Knowledge UNDERSTAND MCQ_SINGLE 1 Mark(s) 2024 AISSCE(Board Exam)
#669
The distance of point \(P(a,b,c)\) from y-axis is :
(A) b
(B) \(b^{2}\)
(C) \(\sqrt{a^{2}+c^{2}}\)
(D) \(a^{2}+c^{2}\)
EASY
6 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2024 AISSCE(Board Exam)
#668
The coordinates of the foot of the perpendicular drawn from the point \((0, 1, 2)\) on the x-axis are given by:
(A) \((1,0,0)\)
(B) \((2,0,0)\)
(C) \((\sqrt{5},0,0)\)
(D) \((0,0,0)\)
MEDIUM
7 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2024 AISSCE(Board Exam)
#667
Direction ratios of a vector parallel to line \(\frac{x-1}{2}=-y=\frac{2z+1}{6}\) are:
(A) \(2,-1,6\)
(B) \(2, 1, 6\)
(C) \(2, 1, 3\)
(D) \(2,-1, 3\)
MEDIUM
8 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2024 AISSCE(Board Exam)
#666
If a line makes an angle of \(30^{\circ}\) with the positive direction of x-axis, \(120^{\circ}\) with the positive direction of y-axis, then the angle which it makes with the positive direction of z-axis is:
(A) \(90^{\circ}\)
(B) \(120^{\circ}\)
(C) \(60^{\circ}\)
(D) \(0^{\circ}\)
MEDIUM
9 Knowledge UNDERSTAND MCQ_SINGLE 1 Mark(s) 2024 AISSCE(Board Exam)
#665
If \(\alpha\), \(\beta\) and \(\gamma\) are the angles which a line makes with positive directions of x, y and z axes respectively, then which of the following is not true?
(A) \(cos^{2}\alpha+cos^{2}\beta+cos^{2}\gamma=1\)
(B) \(sin^{2}\alpha+sin^{2}\beta+sin^{2}\gamma=2\)
(C) \(cos~2\alpha+cos~2\beta+cos~2\gamma=-1\)
(D) \(cos~\alpha+cos~\beta+cos~\gamma=1\)
MEDIUM
10 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2024 AISSCE(Board Exam)
#664
If a line makes an angle of \(\frac{\pi}{4}\) with the positive directions of both x-axis and z-axis, then the angle which it makes with the positive direction of y-axis is:
(A) 0
(B) \(\frac{\pi}{4}\)
(C) \(\frac{\pi}{2}\)
(D) \(\pi\)
MEDIUM
11 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2024 AISSCE(Board Exam)
#663
The vector equation of a line passing through the point (1, -1, 0) and parallel to Y-axis is :
(A) \(\vec{r}=\hat{i}-\hat{j}+\lambda(\hat{i}-\hat{j})\)
(B) \(\vec{r}=\hat{i}-\hat{j}+\lambda\hat{j}\)
(C) \(\vec{r}=\hat{i}-\hat{j}+\lambda\hat{k}\)
(D) \(\vec{r}=\lambda\hat{j}\)
MEDIUM
12 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2024 AISSCE(Board Exam)
#662
The lines \(\frac{1-x}{2}=\frac{y-1}{3}=\frac{z}{1}\) and \(\frac{2x-3}{2p}=\frac{y}{-1}=\frac{z-4}{7}\) are perpendicular to each other for p equal to:
(A) \(-\frac{1}{2}\)
(B) \(\frac{1}{2}\)
(C) 2
(D) 3
MEDIUM
13 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2024 AISSCE(Board Exam)
#661
The angle which the line \(\frac{x}{1}=\frac{y}{-1}=\frac{z}{0}\) makes with the positive direction of Y-axis is:
(A) \(\frac{5\pi}{6}\)
(B) \(\frac{3\pi}{4}\)
(C) \(\frac{5\pi}{4}\)
(D) \(\frac{7\pi}{4}\)
MEDIUM
14 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2024 AISSCE(Board Exam)
#660
The Cartesian equation of the line passing through the point (1, -3, 2) and parallel to the line: \(\vec{r}=(2+\lambda)\hat{i}+\lambda\hat{j}+(2\lambda-1)\hat{k}\) is:
(A) \(\frac{x-1}{2}=\frac{y+3}{0}=\frac{z-2}{-1}\)
(B) \(\frac{x+1}{1}=\frac{y-3}{1}=\frac{z+2}{2}\)
(C) \(\frac{x+1}{2}=\frac{y-3}{0}=\frac{z+2}{-1}\)
(D) \(\frac{x-1}{1}=\frac{y+3}{1}=\frac{z-2}{2}\)
MEDIUM
15 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2023
#826
The value of $\lambda$ for which the angle between the lines $\vec{r}=\hat{i}+\hat{j}+\hat{k}+p(2\hat{i}+\hat{j}+2\hat{k})$ and $\vec{r}=(1+q)\hat{i}+(1+q\lambda)\hat{j}+(1+q)\hat{k}$ is $\frac{\pi}{2}$ :
(A) $-4$
(B) 4
(C) 2
(D) $-2$
MEDIUM
16 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2023
#825
If the direction cosines of a line are $(\frac{1}{a}, \frac{1}{a}, \frac{1}{a})$ then:
(A) $0<a<1$
(B) $a>2$
(C) $a>0$
(D) $a=\pm\sqrt{3}$
MEDIUM
17 Knowledge REMEMBER MCQ_SINGLE 1 Mark(s) 2023
#824
14. Distance of the point $(p, q, r)$ from y-axis is :
(A) q
(B) q
(C) $|q|+|r|$
(D) $\sqrt{p^{2}+r^{2}}$
EASY
18 Knowledge UNDERSTAND MCQ_SINGLE 1 Mark(s) 2023
#823
Direction cosines of the line \frac{x-1}{2}=\frac{1-y}{3}=\frac{2z-1}{12} are:
(A) \frac{2}{7},\frac{3}{7},\frac{6}{7}
(B) \frac{2}{\sqrt{157}},-\frac{3}{\sqrt{157}},\frac{12}{\sqrt{157}}
(C) \frac{2}{7},-\frac{3}{7},-\frac{6}{7}
(D) \frac{2}{7},-\frac{3}{7},\frac{6}{7}
MEDIUM
19 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2023
#822
The angle between the lines 2x=3y=-z and 6x=-y=-4z is
(A) 0°
(B) 30°
(C) 45°
(D) 90°
MEDIUM
20 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2023
#821
If a line makes angles of 90°, 135° and 45° with the x, y and z axes respectively, then its direction cosines are
(A) 0, -1/√2, 1/√2
(B) -1/√2, 0, 1/√2
(C) 1/√2, 0, -1/√2
(D) 0, 1/√2, 1/√2
MEDIUM

Very Short Answer (2 Mark)

1 Knowledge APPLY VSA 2 Mark(s) 2023
#914
Position vectors of the points A, B and C as shown in the figure below are a, $\vec{b}$ and $\vec{c}$ respectively. If $\vec{AC}=\frac{5}{4}\vec{AB}$ , express $\vec{c}$ in terms of $\vec{a}$ and $\vec{b}$ .
OR Check whether the lines given by equations $x=2\lambda+2$, $y=7\lambda+1$, $z=-3\lambda-3$ and $x=-\mu-2,$ $y=2\mu+8,$ $z=4\mu+5$ are perpendicular to each other or not.
MEDIUM
2 Knowledge APPLY VSA 2 Mark(s) 2023
#885
If the angle between the lines $\frac{x-5}{\alpha}=\frac{y+2}{-5}=\frac{z+\frac{24}{5}}{\beta}$ and $\frac{x}{1}=\frac{y}{0}=\frac{z}{1}$ is $\frac{\pi}{4}$, find the relation between $\alpha$ and $\beta$.
MEDIUM
3 Competency APPLY VSA 2 Mark(s) 2023
#882
25. (b) OR: The equations of a line are $5x-3=15y+7=3-10z$. Write the direction cosines of the line and find the coordinates of a point through which it passes.
HARD
4 Knowledge APPLY VSA 2 Mark(s) 2023
#881
25. (a) Find the vector equation of the line passing through the point $(2, 1, 3)$ and perpendicular to both the lines $\frac{x-1}{1}=\frac{y-2}{2}=\frac{z-3}{3} ; \frac{x}{-3}=\frac{y}{2}=\frac{z}{5}$
MEDIUM
5 Knowledge APPLY VSA 2 Mark(s) 2023
#877
Find the vector and the cartesian equations of a line that passes through the point A(1,2,-1) and parallel to the line 5x-25=14-7y=35z.
MEDIUM

Short Answer (3 Marks)

1 Knowledge APPLY SA 3 Mark(s) 2023
#879
Find the distance between the lines: →r = (î + 2î - 4ê) + λ(2î + 3î + 6ê); →r = (3î + 3î - 5ê) + μ(4î + 6î + 12ê)
MEDIUM
2 Knowledge APPLY SA 3 Mark(s) 2023
#878
(a) Find the coordinates of the foot of the perpendicular drawn from the point P(0,2,3) to the line (x+3)/5 = (y-1)/2 = (z+4)/3. OR (b) Three vectors →a, →b and →c satisfy the condition →a + →b + →c = →0. Evaluate the quantity μ = →a·→b + →b·→c + →c·→a, if |→a|=3, |→b|=4 and |→c|=2.
MEDIUM

Long Answer (5 Marks)

1 Competency APPLY LA 5 Mark(s) 2023
#887
Find the equations of all the sides of the parallelogram ABCD whose vertices are $A(4,7,8), B(2,3,4), C(-1,-2,1)$ and $D(1,2,5)$. Also, find the coordinates of the foot of the perpendicular from A to CD.
HARD
2 Competency APPLY LA 5 Mark(s) 2023
#886
Find the value of $b$ so that the lines $\frac{x-1}{2}=\frac{y-b}{3}=\frac{z-3}{4}$ and $\frac{x-4}{5}=\frac{y-1}{2}=z$ are intersecting lines. Also, find the point of intersection of these given lines.
HARD
3 Competency APPLY LA 5 Mark(s) 2023
#884
35. (b) OR: Find the angle between the lines $2x=3y=-z$ and $6x=-y=-4z$.
MEDIUM
4 Competency APPLY LA 5 Mark(s) 2023
#883
35. (a) Show that the following lines do not intersect each other : $\frac{x-1}{3}=\frac{y+1}{2}=\frac{z-1}{5};\frac{x+2}{4}=\frac{y-1}{3}=\frac{z+1}{-2}$
HARD
5 Competency APPLY LA 5 Mark(s) 2023
#880
Find the vector and the Cartesian equations of a line passing through the point (1,2,-4) and parallel to the line joining the points A(3,3,-5) and B(1,0,-11). Hence, find the distance between the two lines. OR Find the equations of the line passing through the points A(1,2,3) and B(3,5,9). Hence, find the coordinates of the points on this line which are at a distance of 14 units from point B.
HARD