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Class 12 Mathematics: 3-D (Three Dimensional) Geometry
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MCQs on direction cosines and line ratios, line equation in space, angle and shortest distance b/w two lines, plane, angle b/w two planes, two lines coplanarity, point distance from plane, angle b/w line and plane.

Class 12 Mathematics: 3-D (Three Dimensional) Geometry
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25 Questions

1. If a, b, c are the direction ratios of the line and l, m, n are the direction cosines of the line, then which of the following is incorrect?
2. Which of the following is the correct formula for the angle between two planes?
3. Find the shortest distance between the following set of parallel lines.
\(\vec{r}=6\hat{i}+2\hat{j}-\hat{k}+λ(\hat{i}+2\hat{j}-4\hat{k})\)
\(\vec{r}=\hat{i}+\hat{j}+\hat{k}+μ(\hat{i}+2\hat{j}-4\hat{k})\)
4. If two lines L1 and L2 are having direction cosines \(l_1,m_1,n_1 \,and \,l_2,m_2,n_2\) respectively, then what is the angle between the two lines?
5. Find the shortest distance between two lines l1 and l2 whose vector equations is given below.
\(\vec{r}=3\hat{i}-4\hat{j}+2\hat{k}+λ(4\hat{i}+\hat{j}+\hat{k})\)
\(\vec{r}=5\hat{i}+\hat{j}-\hat{k}+μ(2\hat{i}-\hat{j}-3\hat{k})\)
6. What is the area of the triangle whose vertices are (0,1), (0,2), (1,5)?
7. If two vectors \(\vec{r}.\vec{n_1}=d_1\) and \(\vec{r}.\vec{n_2}=d_2\) are such that \(\vec{n_1}.\vec{n_2}\)=0, then which of the following is true?
8. Find the vector equation of the line which is passing through the point (1, -4, 4) and parallel to the vector \(2\hat{i}-5\hat{j}+2\hat{k}\).
9. Find the angle between the planes 6x-3y+7z=8 and 2x+3y-2z=5?
10. Find the equation of the line joining A(2,1) and B(6,3) using determinants.
11. Find the vector equation of a line passing through two points (1,0,4) and (6,-3,1).
12. If a line has direction ratios 2, -3, 7 then find the direction cosines.
13. Which of the following is the formula for finding the area of a triangle with the vertices (x1,y1), (x2,y2), (x3,y3).
14. Find the distance of the plane 3x+4y-5z-7=0.
15. Find the equation of the plane passing through the three points (2,2,0), (1,2,1), (-1,2,-2).
16. Find the cartesian equation of the line which is passing through the point (5,-6,1) and parallel to the vector \(5\hat{i}+2\hat{j}-3\hat{k}\).
17. What is the relation between the planes a1x + b1y + c1z + d1 = 0 and a2x + b2y + c21z + d2 = 0, if their normal are perpendicular to each other?
18. If the planes \(A_1 x+B_1 y+C_1 z+D_1\)=0 and \(A_2 x+B_2 y+C_2 z+D_2\)=0 are at right angles to each other, then which of the following is true?
19. If L1 and L2 have the direction ratios \(a_1,b_1,c_1 \,and \,a_2,b_2,c_2\) respectively then what is the angle between the lines?
20. Find the angle between the lines.
\(\frac{x+2}{1}=\frac{y+5}{6}=\frac{z-3}{2}\)
\(\frac{x-4}{5}=\frac{y-3}{-2}=\frac{z+3}{1}\)
21. _____ is the complement of the angle between the line L and a normal line to the plane π.
22. Find the equation between the two parallel lines l1 and l2 whose equations is given below.
\(\vec{r}=3\hat{i}+2\hat{j}-\hat{k}+λ(3\hat{i}-2\hat{j}+\hat{k})\)
\(\vec{r}=2\hat{i}-\hat{j}+\hat{k}+μ(3\hat{i}-2\hat{j}+\hat{k})\)
23. Which of the following is the correct formula for the angle between two planes \(A_1 x+B_1 y+C_1 z+D_1\)=0 and \(A_2 x+B_2 y+C_2 z+D_2\)=0?
24. If the plane passes through three collinear points \((x_1,y_1,z_1),(x_2,y_2,z_2),(x_3,y_3,z_3)\) then which of the following is true?
25. Find the equation of the plane passing through three points (1,2,-1), (0,-1,2) and (3,1,1).

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