Skip to content
← Mathematics

Mathematics · Class 12 Science

Ch 6Line and Plane — Class 12 Mathematics, concept-first.

A line in space is completely determined once we know a point on it together with its direction -- and, equally, two distinct points on a line already fix the direction between them (the segment joining them cannot lie along any other line).

145

Q&A

21

Concepts

10m

Exam weightage

Start learning — read this chapter →

Key concepts

Hover a concept to preview it and jump to its most relevant Q&A.

Vector Equation of a Line through a Point and Parallel to a Vector

A line in space is fully determined once we know one point on it and the direction it runs in. If is a known point with position vector , and is a vector parallel to the line, then any other point on the line satisfies f…

Most relevant Q&A

Start with this concept →

In previous exams

How often this chapter’s concepts have been examined — real appearance data, never estimated.

MHT-CET

1 concept · 1 appearance

2026

Chapter contents

The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.

Introduction

A line in space is completely determined once we know a point on it together with its direction -- and, equally, two distinct points on a line already fix the direction between them (the segment joini…

6.1

Vector and Cartesian equations of a line

A straight line extends infinitely in two opposite directions, so no single equation of the kind we meet with curves in a plane can describe it in three dimensions directly the way describes a line in…

+Exercise 6.1i11 questions
  1. Q1Find the vector equation of the line passing through the point having position vector $-2\hat{i} + \hat{j} + \hat{k}$ and parallel to vector…Free
  2. Q2Find the vector equation of the line passing through points having position vectors $3\hat{i} + 4\hat{j} - 7\hat{k}$ and $6\hat{i} - \hat{j}…Free
  3. Q3Find the vector equation of line passing through the point having position vector $5\hat{i} + 4\hat{j} + 3\hat{k}$ and having direction rati…Free
  4. Q4Find the vector equation of the line passing through the point having position vector $\hat{i} + 2\hat{j} + 3\hat{k}$ and perpendicular to v…Preview
  5. Q5Find the vector equation of the line passing through the point having position vector $-\hat{i} - \hat{j} + 2\hat{k}$ and parallel to the li…Preview
  6. Q6Find the Cartesian equations of the line passing through $A(-1, 2, 1)$ and having direction ratios $2, 3, 1$.Preview
  7. Q7Find the Cartesian equations of the line passing through $A(2, 2, 1)$ and $B(1, 3, 0)$.Preview
  8. Q8$A(-2, 3, 4)$, $B(1, 1, 2)$ and $C(4, -1, 0)$ are three points. Find the Cartesian equations of the line AB and show that points A, B, C are…Preview
  9. Q9Show that lines $\dfrac{x+1}{-10} = \dfrac{y+3}{-1} = \dfrac{z-4}{1}$ and $\dfrac{x+10}{-1} = \dfrac{y+1}{-3} = \dfrac{z-1}{4}$ intersect ea…Preview
  10. Q10A line passes through $(3, -1, 2)$ and is perpendicular to lines $\bar{r} = (\hat{i} + \hat{j} - \hat{k}) + \lambda(2\hat{i} - 2\hat{j} + \h…Preview
  11. Q11Show that the line $\dfrac{x-2}{1} = \dfrac{y-4}{2} = \dfrac{z+4}{-2}$ passes through the origin.Preview
6.1.1

Passing through a point and parallel to a vector

Theorem 6.1 (vector form, point + direction vector). Suppose a line passes through a fixed point whose position vector is , and is parallel to a given vector .

6.1.2

Passing through two points

Theorem 6.3 (vector form, two points). Suppose a line passes through two distinct points and , with position vectors and . Let , with position vector , be any other point on .

6.2

Distance of a point from a line

Theorem 6.5 (distance of a point from a line). Given a line and any point with position vector , we want a formula for the perpendicular distance of from the line, without having to construct the actu…

+Exercise 6.2i9 questions
  1. Q12Find the length of the perpendicular from $(2, -3, 1)$ to the line $\dfrac{x+1}{2} = \dfrac{y-3}{3} = \dfrac{z+1}{-1}$Free
  2. Q13Find the co-ordinates of the foot of the perpendicular drawn from the point $2\hat{i} - \hat{j} + 5\hat{k}$ to the line $\vec{r} = (11\hat{i…Free
  3. Q14Find the shortest distance between the lines $\vec{r} = (4\hat{i} - \hat{j}) + \lambda(\hat{i} + 2\hat{j} - 3\hat{k})$ and $\vec{r} = (\hat{…Free
  4. Q15Find the shortest distance between the lines $\dfrac{x+1}{7} = \dfrac{y+1}{-6} = \dfrac{z+1}{1}$ and $\dfrac{x-3}{1} = \dfrac{y-5}{-2} = \df…Preview
  5. Q16Find the perpendicular distance of the point $(1, 0, 0)$ from the line $\dfrac{x-1}{2} = \dfrac{y+1}{-3} = \dfrac{z+10}{8}$. Also find the c…Preview
  6. Q17A$(1, 0, 4)$, B$(0, -11, 13)$, C$(2, -3, 1)$ are three points and D is the foot of the perpendicular from A to BC. Find the co-ordinates of…Preview
  7. Q18By computing the shortest distance, determine whether following lines intersect each other. $\vec{r} = (\hat{i} - \hat{j}) + \lambda(2\hat{i…Preview
  8. Q19By computing the shortest distance, determine whether following lines intersect each other. $\dfrac{x-5}{4} = \dfrac{y-7}{-5} = \dfrac{z+3}{…Preview
  9. Q20If lines $\dfrac{x-1}{2} = \dfrac{y+1}{3} = \dfrac{z-1}{4}$ and $\dfrac{x-3}{1} = \dfrac{y-k}{2} = \dfrac{z}{1}$ intersect each other then f…Preview
6.3

Skew lines

When two lines lie in space, there are exactly three possible relationships between them. If the two lines meet at a common point, the shortest distance between them is zero.

+Exercise 6.3i11 questions
  1. Q44Find the vector equation of a plane which is at 42 unit distance from the origin and which is normal to the vector $2\hat{i} + \hat{j} - 2\h…Free
  2. Q45Find the perpendicular distance of the origin from the plane $6x - 2y + 3z - 7 = 0$.Free
  3. Q46Find the coordinates of the foot of the perpendicular drawn from the origin to the plane $2x + 6y - 3z = 63$.Free
  4. Q47Reduce the equation $\vec{r} \cdot (3\hat{i} + 4\hat{j} + 12\hat{k}) = 78$ to normal form and hence find (i) the length of the perpendicular…Preview
  5. Q48Find the vector equation of the plane passing through the point having position vector $\hat{i} + \hat{j} + \hat{k}$ and perpendicular to th…Preview
  6. Q49Find the Cartesian equation of the plane passing through A( -1, 2, 3), the direction ratios of whose normal are 0, 2, 5.Preview
  7. Q50Find the Cartesian equation of the plane passing through A(7, 8, 6) and parallel to the XY plane.Preview
  8. Q51The foot of the perpendicular drawn from the origin to a plane is M(1,0,0). Find the vector equation of the plane.Preview
  9. Q52Find the vector equation of the plane passing through the point A(–2, 7, 5) and parallel to vectors $4\hat{i} - \hat{j} + 3\hat{k}$ and $\ha…Preview
  10. Q53Find the Cartesian equation of the plane $\vec{r} = (5\hat{i} - 2\hat{j} - 3\hat{k}) + \lambda(\hat{i} + \hat{j} + \hat{k}) + \mu(\hat{i} -…Preview
  11. Q54Find the vector equation of the plane which makes intercepts 1, 1, 1 on the co-ordinates axes.Preview
6.3.1

Distance between skew lines

Theorem 6.6 (Distance between skew lines). If line has vector equation and line has vector equation , then the shortest distance between and is

6.3.2

Distance between parallel lines

Theorem 6.7 (Distance between parallel lines). The distance between parallel lines and (note that both lines share the same direction vector , which is exactly what makes them parallel) is , where is…

6.4

Equations of Plane

A plane is a flat surface with the property that the line joining any two of its points lies entirely on it.

+Exercise 6.4i5 questions
  1. Q55Find the angle between planes $\vec{r}\cdot(\hat{i}+\hat{j}+2\hat{k})=13$ and $\vec{r}\cdot(2\hat{i}-\hat{j}+\hat{k})=31$.Free
  2. Q56Find the acute angle between the line $\vec{r}=(\hat{i}+2\hat{j}+2\hat{k})+\lambda(2\hat{i}+3\hat{j}-6\hat{k})$ and the plane $\vec{r}\cdot(…Free
  3. Q57Show that lines $\vec{r}=(2\hat{j}-3\hat{k})+\lambda(\hat{i}+2\hat{j}+3\hat{k})$ and $\vec{r}=(2\hat{i}+6\hat{j}+3\hat{k})+\mu(2\hat{i}+3\ha…Preview
  4. Q58Find the distance of the point $4\hat{i}-3\hat{j}+\hat{k}$ from the plane $\vec{r}\cdot(2\hat{i}+3\hat{j}-6\hat{k})=21$.Preview
  5. Q59Find the distance of the point (1, 1, -1) from the plane $3x+4y-12z+20=0$.Preview
6.4.1

Equation of Plane Passing through a Point and Perpendicular to a Vector

Theorem 6.8 (vector form). The plane through the point that is perpendicular to a fixed non-zero vector has the equation , where denotes an arbitrary point of the plane.

6.4.2

The Vector Equation of the Plane Passing through Point A(a-bar) and Parallel to b-bar and c-bar

Theorem 6.10. The plane through the point that is parallel to two given non-zero, non-parallel vectors and has the equation , equivalently .

6.4.3

The Vector Equation of Plane Passing through Three Non-collinear Points

Theorem 6.11. The plane through three non-collinear points , and has the equation .

6.4.4

The Normal Form of Equation of Plane

Theorem 6.12 (normal form). If is the (non-negative) distance of a plane from the origin and is the unit vector along the perpendicular dropped from the origin onto the plane, the plane's equation is…

6.4.5

Equation of Plane Passing through the Intersection of Two Planes

If two planes and actually intersect (i.e. their normals are not parallel), then for every real value of the equation represents some plane passing through their common line of intersection.

6.5

Angle between Planes

So far the angle between two lines has been found by comparing their direction vectors. A plane, unlike a line, does not carry a single direction vector of its own — instead it is characterised by a n…

6.5.1

Angle between Two Planes

Consider two planes and , with normals and . The inclination of the two planes to each other is completely decided by the inclination of and — if the normals are perpendicular, the planes are perpendi…

6.5.2

Angle between a Line and a Plane

Let the line be and the plane be . Two extreme relationships are worth naming first. The line is perpendicular to the plane exactly when its direction is parallel to the plane's normal , i.e.

6.6

Coplanarity of Two Lines

Two lines in space may or may not lie in a common plane. Parallel lines always do — any two parallel lines determine a plane between them.

6.7

Distance of a Point from a Plane

The distance of the origin from a plane already written in normal form is simply — this is the very meaning of the normal form (Fig. 6.13).

Sample & Board Papers

Sample papers and previous-year board questions for this subject.

+Show 45 questions45 questions
  1. Q1Find the vector equation of the plane passing through a point having position vector $3\hat{i} - 2\hat{j} + \hat{k}$ and perpendicular to th…Preview
  2. Q2If the lines $\dfrac{x-1}{-3} = \dfrac{y-2}{2k} = \dfrac{z-3}{2}$ and $\dfrac{x-1}{3k} = \dfrac{y-5}{1} = \dfrac{z-6}{-5}$ are at right angl…Preview
  3. Q3Find the shortest distance between the lines $\bar r = (4\hat i - \hat j) + \lambda(\hat i + 2\hat j - 3\hat k)$ and $\bar r = (\hat i - \ha…Preview
  4. Q4If a line drawn from the point $A(1, 2, 1)$ is perpendicular to the line joining $P(1, 4, 6)$ and $Q(5, 4, 4)$ then find the co-ordinates of…Preview
  5. Q5Find the vector equation of the plane passing through the points $\hat i + \hat j - 2\hat k$, $\hat i + 2\hat j + \hat k$, $2\hat i - \hat j…Preview
  6. Q6If from a point $Q(a, b, c)$ perpendiculars QA and QB are drawn to the YZ and ZX planes respectively, then find the vector equation of the p…Preview
  7. Q7Find the cartesian equation of the line passing throught the points $A(3, 4, -7)$ and $B(6, -1, 1)$.Preview
  8. Q8Find the shortest distance between the lines $\dfrac{x-1}{2} = \dfrac{y-2}{3} = \dfrac{z-3}{4}$ and $\dfrac{x-2}{3} = \dfrac{y-4}{4} = \dfra…Preview
  9. Q9Find the vector and cartesian equations of the plane passing through the points $A(1, 1, -2)$, $B(1, 2, 1)$ and $C(2, -1, 1)$.Preview
  10. Q10The measure of acute angle between the lines whose direction ratios are 3, 2, 6 and $-2$, 1, 2 is ______. (a) $\cos^{-1}\left(\dfrac{1}{7}\r…Preview
  11. Q11If a line makes angles $\alpha, \beta, \gamma$ with the co-ordinate axes, prove that $\cos 2\alpha + \cos 2\beta + \cos 2\gamma + 1 = 0$.Preview
  12. Q12Find the distance of the point $(1, 2, -1)$ from the plane $x - 2y + 4z - 10 = 0$.Preview
  13. Q13Find the vector equation of the line which passes through the point with position vector $4\hat{i} - \hat{j} + 2\hat{k}$ and is in the direc…Preview
  14. Q14If the origin is the centroid of the triangle whose vertices are $A(2, p, -3)$, $B(q, -2, 5)$ and $R(-5, 1, r)$, then find the values of $p,…Preview
  15. Q15Find the angle between the lines $\dfrac{x-1}{4} = \dfrac{y-3}{1} = \dfrac{z}{8}$ and $\dfrac{x-2}{2} = \dfrac{y+1}{2} = \dfrac{z-4}{1}$.Preview
  16. Q16Find the vector equation of the plane passing through the points $A(1, 0, 1)$, $B(1, -1, 1)$ and $C(4, -3, 2)$.Preview
  17. Q17The acute angle between the two planes $x + y + 2z = 3$ and $3x - 2y + 2z = 7$ is ________. (a) $\sin^{-1}\left(\dfrac{5}{\sqrt{102}}\right)…Preview
  18. Q18The direction ratios of the line which is perpendicular to the lines with direction ratios $-1, 2, 2$ and $0, 2, 1$ are ________. (a) $-2, -…Preview
  19. Q19Write the equation of the plane $3x + 4y - 2z = 5$ in the vector form.Preview
  20. Q20The equation of a line is $2x - 2 = 3y + 1 = 6z - 2$, find its direction ratios and also find the vector equation of the line.Preview
  21. Q21The cartesian equation of the line passing through the points A(4, 2, 1) and B(2, -1, 3) is ________. (a) $\dfrac{x+4}{2}=\dfrac{y-2}{3}=\df…Preview
  22. Q22If the line $\bar r = (\hat i - 2\hat j + 3\hat k) + \lambda(2\hat i + \hat j + 2\hat k)$ is parallel to the plane $\bar r \cdot (3\hat i -…Preview
  23. Q23Find the equation of the line passing through the point (3, 1, 2) and perpendicular to the lines $\dfrac{x-1}{1}=\dfrac{y-2}{2}=\dfrac{z-3}{…Preview
  24. Q24Find the distance of the point $\hat i + 2\hat j - \hat k$ from the plane $\bar r \cdot (\hat i - 2\hat j + 4\hat k) = 10$Preview
  25. Q25Equation of line passing through the points (0, 0, 0) and (2, 1, -3) is ________. (a) $\dfrac{x}{2}=\dfrac{y}{1}=\dfrac{z}{-3}$ (b) $\dfrac{…Preview
  26. Q26Find the cartesian equation of the plane passing through A(1, 2, 3) and the direction ratios of whose normal are 3, 2, 5.Preview
  27. Q27Find the distance between the parallel lines $\dfrac{x}{2}=\dfrac{y}{-1}=\dfrac{z}{2}$ and $\dfrac{x-1}{2}=\dfrac{y-1}{-1}=\dfrac{z-1}{2}$Preview
  28. Q28Find the vector equation of the plane passing through the point A(-1, 2, -5) and parallel to the vectors $4\hat i - \hat j + 3\hat k$ and $\…Preview
  29. Q29Find the shortest distance between lines $\dfrac{x-1}{2}=\dfrac{y-2}{3}=\dfrac{z-3}{4}$ and $\dfrac{x-2}{3}=\dfrac{y-4}{4}=\dfrac{z-5}{5}$Preview
  30. Q30Lines $\bar r = (\hat i + \hat j - \hat k) + \lambda(2\hat i - 2\hat j + \hat k)$ and $\bar r = (4\hat i - 3\hat j + 2\hat k) + \mu(\hat i -…Preview
  31. Q31Find the length of the perpendicular drawn from the point P(3, 2, 1) to the line $\bar r = (7\hat i + 7\hat j + 6\hat k) + \lambda(-2\hat i…Preview
  32. Q32The perpendicular distance of the plane $\bar{r}\cdot(3\hat{i}+4\hat{j}+12\hat{k})=78$, from the origin is ____. (a) 4 (b) 5 (c) 6 (d) 8Preview
  33. Q33Find the vector equation of the line passing through the point having position vector $4\hat{i}-\hat{j}+2\hat{k}$ and parallel to the vector…Preview
  34. Q34Find the shortest distance between the lines $\bar r=(4\hat{i}-\hat{j})+\lambda(\hat{i}+2\hat{j}-3\hat{k})$ and $\bar r=(\hat{i}-\hat{j}-2\h…Preview
  35. Q35Find the angle between the line $\bar r=(\hat{i}+2\hat{j}+\hat{k})+\lambda(\hat{i}+\hat{j}+\hat{k})$ and the plane $\bar r\cdot(2\hat{i}+\ha…Preview
  36. Q36The vector equation of the line passing through the point having position vector $4\hat i - \hat j + 2\hat k$ and parallel to vector $-2\hat…Preview
  37. Q37A line makes angles of measure $45°$ and $60°$ with the positive directions of the $Y$ and $Z$ axes respectively. Find the angle made by the…Preview
  38. Q38Find the vector equation of the plane passing through the point having position vector $2\hat i+3\hat j+4\hat k$ and perpendicular to the ve…Preview
  39. Q39A line passes through the points $(6,-7,-1)$ and $(2,-3,1)$. Find the direction ratios and the direction cosines of the line. Show that the…Preview
  40. Q40Find the cartesian and vector equations of the line passing through $A(1,2,3)$ and having direction ratios $2,3,7$.Preview
  41. Q41Find the vector equation of the plane passing through points $A(1,1,2)$, $B(0,2,3)$ and $C(4,5,6)$.Preview
  42. Q42The angle between the line $\bar r=(\hat i+2\hat j+\hat k)+\lambda(\hat i+\hat j+\hat k)$ and the plane $\bar r\cdot(2\hat i-\hat j+\hat k)=…Preview
  43. Q43Find the vector equation of the line passing through the points $A(1,2,3)$ and $B(2,3,4)$.Preview
  44. Q44Show that the lines $\bar r=(\hat i+\hat j-\hat k)+\lambda(2\hat i-2\hat j+\hat k)$ and $\bar r=(4\hat i-3\hat j+2\hat k)+\mu(\hat i-2\hat j…Preview
  45. Q45Find the cartesian equation of the plane $\bar r=(\hat i-\hat j)+\lambda(\hat i+\hat j+\hat k)+\mu(\hat i-2\hat j+3\hat k)$.Preview

More questions

64 Q
+Show 23 questions23 questions
  1. Q21Find the vector equation of the line passing through the point having position vector $3\hat{i} + 4\hat{j} - 7\hat{k}$ and parallel to $6\ha…Free
  2. Q22Find the vector equation of the line which passes through the point $(3, 2, 1)$ and is parallel to the vector $2\hat{i} + 2\hat{j} - 3\hat{k…Free
  3. Q23Find the Cartesian equations of the line which passes through the point $(-2, 4, -5)$ and parallel to the line $\dfrac{x+2}{3} = \dfrac{y-3}…Free
  4. Q24Obtain the vector equation of the line $\dfrac{x+5}{3} = \dfrac{y+4}{5} = \dfrac{z+5}{6}$.Preview
  5. Q25Find the vector equation of the line which passes through the origin and the point $(5, -2, 3)$.Preview
  6. Q26Find the Cartesian equations of the line which passes through points $(3, -2, -5)$ and $(3, -2, 6)$.Preview
  7. Q27Find the Cartesian equations of the line passing through A$(3, 2, 1)$ and B$(1, 3, 1)$.Preview
  8. Q28Find the Cartesian equations of the line passing through the point A$(1, 1, 2)$ and perpendicular to vectors $\vec{b} = \hat{i} + 2\hat{j} +…Preview
  9. Q29Find the Cartesian equations of the line which passes through the point $(2, 1, 3)$ and perpendicular to lines $\dfrac{x-1}{1} = \dfrac{y-2}…Preview
  10. Q30Find the vector equation of the line which passes through the origin and intersect the line $x - 1 = y - 2 = z - 3$ at right angle.Preview
  11. Q31Find the value of $\lambda$ so that lines $\dfrac{1-x}{3} = \dfrac{7y-14}{2\lambda} = \dfrac{z-3}{2}$ and $\dfrac{7-7x}{3\lambda} = \dfrac{y…Preview
  12. Q32Find the acute angle between lines $\dfrac{x-1}{1} = \dfrac{y-2}{-1} = \dfrac{z-3}{2}$ and $\dfrac{x-1}{2} = \dfrac{y-2}{1} = \dfrac{z-3}{1}…Preview
  13. Q33Find the acute angle between lines $x = y, z = 0$ and $x = 0, z = 0$.Preview
  14. Q34Find the acute angle between lines $x = -y, z = 0$ and $x = 0, z = 0$.Preview
  15. Q35Find the co-ordinates of the foot of the perpendicular drawn from the point $(0, 2, 3)$ to the line $\dfrac{x+3}{5} = \dfrac{y-1}{2} = \dfra…Preview
  16. Q36By computing the shortest distance determine whether following lines intersect each other. $\vec{r} = (\hat{i} + \hat{j} - \hat{k}) + \lambd…Preview
  17. Q37By computing the shortest distance determine whether following lines intersect each other. $\dfrac{x-5}{4} = \dfrac{y-7}{5} = \dfrac{z+3}{5}…Preview
  18. Q38If lines $\dfrac{x-1}{2} = \dfrac{y+1}{3} = \dfrac{z-1}{4}$ and $\dfrac{x-2}{1} = \dfrac{y+m}{2} = \dfrac{z-2}{1}$ intersect each other then…Preview
  19. Q39Find the vector and Cartesian equations of the line passing through the point $(-1, -1, 2)$ and parallel to the line $2x - 2 = 3y + 1 = 6z -…Preview
  20. Q40Find the direction cosines of the line $\vec{r} = \left(-2\hat{i} + \dfrac{5}{2}\hat{j} - \hat{k}\right) + \lambda(2\hat{i} + 3\hat{j})$.Preview
  21. Q41Find the Cartesian equation of the line passing through the origin which is perpendicular to $x - 1 = y - 2 = z - 1$ and intersects the $\df…Preview
  22. Q42Write the vector equation of the line whose Cartesian equations are $y = 2$ and $4x - 3z + 5 = 0$.Preview
  23. Q43Find the co-ordinates of points on the line $\dfrac{x-1}{1} = \dfrac{y-2}{-2} = \dfrac{z-3}{2}$ which are at the distance $3$ unit from the…Preview
+Show 20 questions20 questions
  1. Q60If the line $\dfrac{x}{3}=\dfrac{y}{4}=z$ is perpendicular to the line $\dfrac{x-1}{k}=\dfrac{y+2}{3}=\dfrac{z-3}{k-1}$ then the value of $k…Free
  2. Q61The vector equation of line $2x-1=3y+2=z-2$ is (A) $\vec{r}=\left(\dfrac{1}{2}\hat{i}-\dfrac{2}{3}\hat{j}+2\hat{k}\right)+\lambda(3\hat{i}+2…Free
  3. Q62The direction ratios of the line which is perpendicular to the two lines $\dfrac{x-7}{2}=\dfrac{y+17}{-3}=\dfrac{z-6}{1}$ and $\dfrac{x+5}{1…Free
  4. Q63The length of the perpendicular from (1, 6, 3) to the line $\dfrac{x}{1}=\dfrac{y-1}{2}=\dfrac{z-2}{3}$ is (A) 3 (B) $\sqrt{11}$ (C) $\sqrt{…Preview
  5. Q64The shortest distance between the lines $\vec{r}=(\hat{i}+2\hat{j}+\hat{k})+\lambda(\hat{i}-\hat{j}-\hat{k})$ and $\vec{r}=(2\hat{i}-\hat{j}…Preview
  6. Q65The lines $\dfrac{x-2}{1}=\dfrac{y-3}{1}=\dfrac{z-4}{-k}$ and $\dfrac{x-1}{k}=\dfrac{y-4}{2}=\dfrac{z-5}{1}$ are coplanar if (A) k = 1 or -1…Preview
  7. Q66The lines $\dfrac{x}{1}=\dfrac{y}{2}=\dfrac{z}{3}$ and $\dfrac{x-1}{-2}=\dfrac{y-2}{-4}=\dfrac{z-3}{6}$ are (A) perpendicular (B) inrersecti…Preview
  8. Q67Equation of X-axis is (A) x = y = z (B) y = z (C) y = 0, z = 0 (D) x = 0, y = 0Preview
  9. Q68The angle between the lines $2x=3y=-z$ and $6x=-y=-4z$ is (A) 45° (B) 30° (C) 0° (D) 90°Preview
  10. Q69The direction ratios of the line $3x+1=6y-2=1-z$ are (A) 2, 1, 6 (B) 2, 1, -6 (C) 2, -1, 6 (D) -2, 1, 6Preview
  11. Q70The perpendicular distance of the plane $2x+3y-z=k$ from the origin is $\sqrt{14}$ units, the value of $k$ is (A) 14 (B) 196 (C) $2\sqrt{14}…Preview
  12. Q71The angle between the planes $\vec{r}\cdot(\hat{i}-2\hat{j}+3\hat{k})+4=0$ and $\vec{r}\cdot(2\hat{i}+\hat{j}-3\hat{k})+7=0$ is (A) $\dfrac{…Preview
  13. Q72If the planes $\vec{r}\cdot(2\hat{i}-\lambda\hat{j}+\hat{k})=3$ and $\vec{r}\cdot(4\hat{i}-\hat{j}+\mu\hat{k})=5$ are parallel, then the val…Preview
  14. Q73The equation of the plane passing through (2, -1, 3) and making equal intercepts on the coordinate axes is (A) x + y + z = 1 (B) x + y + z =…Preview
  15. Q74Measure of angle between the planes $5x-2y+3z-7=0$ and $15x-6y+9z+5=0$ is (A) 0° (B) 30° (C) 45° (D) 90°Preview
  16. Q75The direction cosines of the normal to the plane $2x-y+2z=3$ are (A) $\dfrac{2}{3}, \dfrac{-1}{3}, \dfrac{2}{3}$ (B) $\dfrac{-2}{3}, \dfrac{…Preview
  17. Q76The equation of the plane passing through the points (1, -1, 1), (3, 2, 4) and parallel to Y-axis is: (A) 3x + 2z - 1 = 0 (B) 3x - 2z = 1 (C…Preview
  18. Q77The equation of the plane in which the line $\dfrac{x-5}{4}=\dfrac{y-7}{4}=\dfrac{z+3}{-5}$ and $\dfrac{x-8}{7}=\dfrac{y-4}{1}=\dfrac{z+5}{3…Preview
  19. Q78If the line $\dfrac{x+1}{2}=\dfrac{y-m}{3}=\dfrac{z-4}{6}$ lies in the plane $3x-14y+6z+49=0$, then the value of $m$ is: (A) 5 (B) 3 (C) 2 (…Preview
  20. Q79The foot of perpendicular drawn from the point (0,0,0) to the plane is (4, -2, -5) then the equation of the plane is (A) 4x + y + 5z = 14 (B…Preview
+Show 21 questions21 questions
  1. Q80Find the vector equation of the plane which is at a distance of 5 unit from the origin and which is normal to the vector $2\hat{i}+\hat{j}+2…Free
  2. Q81Find the perpendicular distance of the origin from the plane $6x+2y+3z-7=0$.Free
  3. Q82Find the coordinates of the foot of the perpendicular drawn from the origin to the plane $2x+3y+6z=49$.Free
  4. Q83Reduce the equation $\vec{r}\cdot(6\hat{i}+8\hat{j}+24\hat{k})=13$ to normal form and hence find (i) the length of the perpendicular from th…Preview
  5. Q84Find the vector equation of the plane passing through the points A(1, -2, 1), B(2, -1, -3) and C(0, 1, 5).Preview
  6. Q85Find the Cartesian equation of the plane passing through A(1, -2, 3) and the direction ratios of whose normal are 0, 2, 0.Preview
  7. Q86Find the Cartesian equation of the plane passing through A(7, 8, 6) and parallel to the plane $\vec{r}\cdot(6\hat{i}+8\hat{j}+7\hat{k})=0$.Preview
  8. Q87The foot of the perpendicular drawn from the origin to a plane is M(1, 2, 0). Find the vector equation of the plane.Preview
  9. Q88A plane makes non zero intercepts a, b, c on the co-ordinates axes. Show that the vector equation of the plane is $\vec{r}\cdot(bc\hat{i}+ca…Preview
  10. Q89Find the vector equation of the plane passing through the point A(-2, 3, 5) and parallel to vectors $4\hat{i}+3\hat{k}$ and $\hat{i}+\hat{j}…Preview
  11. Q90Find the Cartesian equation of the plane $\vec{r}=\lambda(\hat{i}+\hat{j}-\hat{k})+\mu(\hat{i}+2\hat{j}+3\hat{k})$.Preview
  12. Q91Find the vector equations of planes which pass through A(1, 2, 3), B(3, 2, 1) and make equal intercepts on the co-ordinates axes.Preview
  13. Q92Find the vector equation of the plane which makes equal non-zero intercepts on the co-ordinates axes and passes through (1,1,1).Preview
  14. Q93Find the angle between planes $\vec{r}\cdot(-2\hat{i}+\hat{j}+2\hat{k})=17$ and $\vec{r}\cdot(2\hat{i}+2\hat{j}+\hat{k})=71$.Preview
  15. Q94Find the acute angle between the line $\vec{r}=\lambda(\hat{i}-\hat{j}+\hat{k})$ and the plane $\vec{r}\cdot(2\hat{i}-\hat{j}+\hat{k})=23$.Preview
  16. Q95Show that lines $\vec{r}=(\hat{i}+4\hat{j})+\lambda(\hat{i}+2\hat{j}+3\hat{k})$ and $\vec{r}=(3\hat{j}-\hat{k})+\mu(2\hat{i}+3\hat{j}+4\hat{…Preview
  17. Q96Find the distance of the point $3\hat{i}+3\hat{j}+\hat{k}$ from the plane $\vec{r}\cdot(2\hat{i}+3\hat{j}+6\hat{k})=21$.Preview
  18. Q97Find the distance of the point (13, 13, -13) from the plane $3x+4y-12z=0$.Preview
  19. Q98Find the vector equation of the plane passing through the origin and containing the line $\vec{r}=(\hat{i}+4\hat{j}+\hat{k})+\lambda(\hat{i}…Preview
  20. Q99Find the vector equation of the plane which bisects the segment joining A(2,3,6) and B(4,3,-2) at right angle.Preview
  21. Q100Show that lines x = y, z = 0 and x + y = 0, z = 0 intersect each other. Find the vector equation of the plane determined by them.Preview