1

THREE DIMENSIONAL GEOMETRY CLASS 12

DIRECTION COSINES OF A LINE (DC’S)

If a line makes angles 𝛼, 𝛽, 𝛾 with the +ve direction of x, y, z axis respectively, then DC’S of the line is 𝑙 = 𝑐𝑜𝑠𝛼, 𝑚 = 𝑐𝑜𝑠𝛽, 𝑛 = 𝑐𝑜𝑠𝛾

𝑙2 + 𝑚2 + 𝑛2 = 1 Co-ordinates of point P are (𝑙𝑟, 𝑚𝑟, 𝑛𝑟) DR’S are proportional to DC’S DR’S of a line joining two points 𝑃(𝑥1, 𝑦1, 𝑧1)𝑎𝑛𝑑 𝑄(𝑥2, 𝑦2, 𝑧2) are

𝑥2 − 𝑥1, 𝑦2 − 𝑦1, 𝑧2 − 𝑧1

DC’S are 𝑥2−𝑥1𝑟

,𝑦2−𝑦1

𝑟,

𝑧2−𝑧1

𝑟 where 𝑟 = √(𝑥2 − 𝑥1)

2 + (𝑦2 − 𝑦1)2 + (𝑧2 − 𝑧1)

2

ANGLE BETWEEN TWO LINES

Angle between two lines whose DR’S are (𝑎1, 𝑏1, 𝑐1)& (𝑎2, 𝑏2 , 𝑐2) is given by 𝑐𝑜𝑠𝜃 =

𝑎1𝑎2+𝑏1𝑏2+𝑐1𝑐2

√𝑎12+𝑏1

2+𝑐12 √𝑎2

2+𝑏22+𝑐2

2

If 𝑎1𝑎2 + 𝑏1𝑏2 + 𝑐1𝑐2 = 0 ⇒ 𝑙𝑖𝑛𝑒𝑠 𝑎𝑟𝑒 ⊥. If 𝑎1

𝑎2=

𝑏1

𝑏2=

𝑐1

𝑐2 ⇒ 𝑙𝑖𝑛𝑒𝑠 𝑎𝑟𝑒 ∥

Angle between two lines whose DC’S are 𝑙1, 𝑚1, 𝑛1 𝑎𝑛𝑑 𝑙2, 𝑚2, 𝑛2 is given by cos 𝜃 = 𝑙1𝑙2 + 𝑚1𝑚2 + 𝑛1𝑛2

If 𝑙1𝑙 + 𝑚1𝑚 + 𝑛1𝑛2 = 0 ⇒ 𝑙𝑖𝑛𝑒𝑠 𝑎𝑟𝑒 ⊥. If 𝑙1 = 𝑙2, 𝑚1 = 𝑚2, 𝑛1 = 𝑛2 ⇒ 𝑙𝑖𝑛𝑒𝑠 𝑎𝑟𝑒 ∥

EXERCISE 1

Q1. If a line makes angles 90°, 60° 𝑎𝑛𝑑 30° with the positive directions of x, y and z- axes respectively,

find its direction cosines. (0, ½, √3

2)

Q2. Can the numbers 1

√2 ,

1

√2,

1

√2 be the direction cosines of a line? (NO)

Q3. Can a line have direction angles 45°, 60°, 120°? (YES)

Q4. Write the direction cosines of x, y and z axis. [ (1,0,0); (0,1,0);(0,0,1)]

Q5. Write the direction cosines of a line equally inclined to the three coordinate axes. (±1

√3, ±

1

√3, ±

1

√3)

Q6. If a line makes angles 𝛼, 𝛽 𝑎𝑛𝑑 𝛾 with the coordinate axes, then find the value of

𝑐𝑜𝑠2𝛼 + 𝑐𝑜𝑠2𝛽 + 𝑐𝑜𝑠2𝛾. [-1]

2

Q7. A line makes an angle of 𝜋

4 with each of x-axis and y-axes. What angle does it make with z-axis?

[𝜋/2]

Q8. A line in XY- plane makes an angle of 30° with the X-axis, find the direction cosines of the line.

[<√3

2,

1

2, 0 >]

Q9. If the direction cosines of the line are 2

3, −

1

3, −

2

3, then find its direction ratios.

[, 𝜆 ≠ 0]

Q10. If a line have direction ratios 2 ,-1, -2, determine its direction cosines. []

Q11. Write the direction cosines of the line joining the points (1,0,0) and (0,1,1). []

Q12. Show that the line joining the origin to the point (2,1,1) is perpendicular to the line determined by

the points (3,5,-1) and (4,3,-1).

Q13. Show that the point through the points (1,-1,2) and (3,4,-2) is perpendicular to the line through the

points (0,3,2) and (3,5,6).

Q14. Show that the line through the points (4,7,8) and (2,3,4) is parallel to the line through the points

(-1,-2,1) and (1,2,5).

Q15. For what value of p will the line through (4,1,2) and (5,p,0) be perpendicular to the line through

(2,1,1) and (3,3,-1). [-3/2]

Q16. Find the value of p and q so that the line joining the points (7,p,2) and (q, -2, 5) may be parallel to

the line joining the points (2, -3, 5) and (-6, -15, 11). [p=4,q=3]

Q17. Show that the points A(2, 3, -4) , B(1,-2, 3) and C(3,8,-11) are collinear.

Q18. Find the angle between the lines whose direction ratios are (1,2,1) and (2,-3,4). [90°]

Q19. If A, B, C are the points (1,4,2), (-2, 1, 2), (2, -3, 4), then find the angles of the triangle ABC. What

type of triangle is it? ∠𝐴 = 𝑐𝑜𝑠−1 (1

√3) ; ∠𝐵 = 90°; ∠𝐶 = 𝑐𝑜𝑠−1√ 2/3]

Q20. If the coordinates of the points A, B, C and D are (1,2,3), (4,5,7), (-4,3,-6) and (2,9,2) respectively,

then find the angle between the lines AB and CD. [0°]

Q21. Show that the points A(4,7,8), B(2,3,4), C(-1,-2,1) and D(1,2,5) taken in order are the vertices of a

parallelogram.

Q22. Find the direction cosines of a line which is perpendicular to the lines formed by joining A(2,3,-4) to

B(-3,3,-2) and C(-1,4,2) to D(3,5,1). []

Q23. If a line L makes angles 𝛼, 𝛽, 𝛾, 𝛿 with the four diagonals of a cube, then prove that

𝑐𝑜𝑠2𝛼 + 𝑐𝑜𝑠2𝛽 + 𝑐𝑜𝑠2𝛾 + 𝑐𝑜𝑠2𝛿 =4

3.

3

Q24. Find the point in which the join of A(-9,4,5) and B(11,0,-1) is met by the perpendicular from the

origin. (1,2,2)

Q25. Find the coordinates of the foot of perpendicular drawn from the point A(1,2,1) on the line joining

the points B(1,4,6) and C(5,4,4). (3,4,5)

STRAIGHT LINE (IN SPACE)

POINT – DIRECTION FORM

(I) CARTESIAN EQUATION Equation of line passing through the points (𝑥1, 𝑦1 , 𝑧1) and having

DR’S (𝑎, 𝑏, 𝑐) is 𝑥−𝑥1

𝑎=

𝑦−𝑦1

𝑏=

𝑧−𝑧1

𝑐

(II) VECTOR EQUATION Equation of a line passing through a point A with position vector �⃗�

and parallel to �⃗⃗� is 𝑟 = �⃗� + 𝜆�⃗⃗�

TWO-POINT FORM

(I) CARTESIAN EQUATION

Equation of a line passing through the points A(𝑥1, 𝑦1, 𝑧1) and

B(𝑥2, 𝑦2, 𝑧2) is 𝑥−𝑥1

𝑥2−𝑥1=

𝑦−𝑦1

𝑦2−𝑦1=

𝑧−𝑧1

𝑧2−𝑧1

(II) VECTOR EQUATION

Equation of a line passing through the points A and B with position

vectors �⃗� and �⃗⃗� is given by 𝑟 = �⃗� + 𝜆(�⃗⃗� − �⃗�)

EXERCISE 2

Q1. Find the vector equation of the line passing through the point (1,2,3) and parallel to the vector 3𝑖 ̂ +

2𝑗̂ − 2𝑘.̂ [𝑟 = 𝑖̂ + 2𝑗̂ + 3�̂� + 𝜆(3𝑖̂ + 2𝑗̂ − 2�̂�)

Q2. Find the Cartesian equation of the line which passes through the point with position vector 2𝑖̂ − 𝑗̂ +

4�̂� and in the direction of the vector 𝑖̂ + 2𝑗̂ − 𝑘.̂ [ 𝑥−2

1=

𝑦+1

2=

𝑧−4

−1 ]

Q3. Write the vector equation of the line 𝑥−5

3=

𝑦+7

7=

𝑧−6

2 [𝑟 = 5𝑖̂ − 4𝑗̂ + 6�̂� + 𝜆(3𝑖̂ + 7𝑗̂ + 2�̂�)]

4

Q4. Write the Cartesian equation of the following line given in vector form

�̂� = 2𝑖̂ + 𝑗̂ − 4�̂� + 𝜆(𝑖̂ − 𝑗̂ − �̂�). [𝑥−2

1=

𝑦−1

−1=

𝑧+4

−1]

Q5. Find the Cartesian equations of the line which passes through the points (-2,4,-5) and parallel to

the line 𝑥+3

3=

𝑦−4

5=

𝑧+8

6. [

𝑥+2

3=

𝑦−4

5=

𝑧+5

6]

Q6. Find the vector equation of the line passing thought he points (-1,0,2) and (3,4,6).

Q7. Find the vector and the Cartesian equations of the line passing through origin and the point

(5,-2,3). [𝑟 = −𝑖̂ + 2𝑗̂ + 𝜆(4𝑖̂ + 4𝑗̂ + 4�̂�)]

Q8. The Cartesian equation of the line AB is 2𝑥−1

2=

4−𝑦

7=

𝑧+1

2. Write the direction ratios of a line parallel

to AB. [(1,-7,2)]

Q9. The Cartesian equation of the line is 3𝑥 − 3 = 2𝑦 + 1 = 5 − 6𝑧. Write the direction ratios of the

line. [(2,3,1)]

Q10. Find the vector equation of x-axes. [𝑟 = 𝜆𝑖̂]

Q11. Find the vector and Cartesian equation of the line through the point (5,2,-4) and which is parallel

to the vector 3𝑖̂ + 2𝑗̂ − 8�̂�. [𝑥−5

3=

𝑦−2

2=

𝑧+4

−8, 𝑟 = 5𝑖̂ + 2𝑗̂ − 4�̂� + 𝜆(3𝑖̂ + 2𝑗̂ − 8�̂�)]

Q12. A line passes through the point with position vector 2𝑖̂ − 𝑗̂ + 4�̂� and is in the direction of 𝑖̂ + 𝑗̂ −

2�̂�. Find the equation of the line in vector and Cartesian form. [𝑥−2

1=

𝑦+1

1=

𝑧−4

−2]

Q13. Find the vector equation of the line with position vector 2𝑖̂ − 𝑗̂ + 4 �̂� and in the direction of î + ĵ

− 2 k̂. Find the equation of the line in vector and in Cartesian form.

Q14. Find the vector equation of a line passing through the point with position vector 𝑖̂ − 2 𝑗̂ − 3 �̂� and

parallel to the line joining the points with position vectors 𝑖̂ − 𝑗̂ + 4�̂� and 2𝑖̂ + 𝑗̂ + 2�̂�. Also find the

Cartesian equation of the line. [𝑟 = 𝑖̂ − 2𝑗̂ − 3�̂� + 𝜆(𝑖̂ + 2𝑗̂ − 2�̂�),𝑥−1

1=

𝑦+2

2=

𝑧+3

−2]

Q15. The Cartesian equation of a line is 6𝑥 − 2 = 3𝑦 + 1 = 2𝑧 − 2. find direction ratios of the line and

write down the vector equation of the line through (2, −1, −1) which is parallel to the given line.

[(1,2,3); 𝑟 = 2𝑖̂ − 𝑗̂ − �̂� + 𝜆(𝑖̂ + 2𝑗̂ + 3�̂�)]

Q16. Find the equation of a line through 𝐴(1, −1,5) and parallel to the line 𝑥−2

3=

𝑦−5

−2, 𝑧 = −1.

[𝑥−1

3=

𝑦+1

−2; 𝑧 = 5]

Q17. The points A(4,5,10), B(2,3,4)and C(1,2, −1) are three vertices of a parallelogram ABCD. Find the

vector equations of the sides AB and BC. Also, find the coordinates of D.

[𝑟 = 4𝑖̂ + 5𝑗̂ + 10�̂� + 𝜆(𝑖̂ + 𝑗̂ + 3�̂�), 𝑟 = 2𝑖̂ + 3𝑗̂ + 4�̂� + 𝜇(𝑖̂ + 𝑗̂ + 5�̂�); 𝐷(3,4,5)]

5

Q18. Find the coordinates of the point where the line through A(3,4,1) and B(5,1,6) crosses the XY-

plane. [(13/5,23/5,0)]

Q19. Find the coordinates of the points on the line 𝑥−1

2=

𝑦+2

3=

𝑧−3

6 which are at a distance of 3 units

from the point (1,-2,3). [(13/7,-5/7,39/7) or (1/7,-23/7)3/7)]

Q20. Find a point on the line 𝑥+2

3=

𝑦+1

2=

𝑧−3

2 at a distance of 3√2 units from the point (1,2,3).

[(-2,-1,3),(56/17,43/17,111,17)]

Q21. Show that the lines 𝑥−5

4=

𝑦−7

4=

𝑧+3

−5 and

𝑥−8

7=

𝑦−4

1=

𝑧−5

3 intersect. Find the point of

intersection. [(1,3,2)]

Q22. Prove that the line through A(0,-1,-1) and B(4,5,1) intersects the line through C(3,9,4) and D(-

4,4,4). Also, find the point of intersection. [(10,14,4)]

Q23. Show that the lInes 𝑥−1

3=

𝑦+1

2=

𝑧−1

5 and

𝑥−2

4=

𝑦−1

3=

𝑧+1

−2 do not intersect.

ANGLE BETWEEN TWO LINES

(I) CARTESIAN EQUATION

𝑥 − 𝑥1𝑎1

=𝑦 − 𝑦1

𝑏1=

𝑧 − 𝑧1𝑐1

𝑥−𝑥2

𝑎2=

𝑦−𝑦2

𝑏2=

𝑧−𝑧2

𝑐2

is given by 𝑐𝑜𝑠𝜃 = 𝑎1𝑎2+𝑏1𝑏2+𝑐1𝑐2

√𝑎12+𝑏12+𝑐12√𝑎22+𝑏2

2+𝑐22

(II) VECTOR FORM

𝑟 = 𝑎1⃗⃗⃗⃗⃗ + λb1⃗⃗⃗⃗⃗

𝑟 = 𝑎2⃗⃗⃗⃗⃗ + λb2⃗⃗⃗⃗⃗

is given by 𝑐𝑜𝑠𝜃 = 𝑏1⃗⃗ ⃗⃗ ⃗ ∙ 𝑏2⃗⃗ ⃗⃗ ⃗

|𝑏1⃗⃗ ⃗⃗ ⃗||𝑏2⃗⃗ ⃗⃗ ⃗|

6

EXERCISE 3

Q1. Find the angle between the following pairs of lines:

i. 𝑟 = 4𝑖̂ − 𝑗̂ + 𝜆(𝑖̂ + 2𝑗̂ − 2�̂�) and 𝑟 = 𝑖̂ − 𝑗̂ + 2�̂� − 𝜇(2𝑖̂ + 4𝑗̂ − 4�̂�) [0°] ii. 𝑟 = 2𝑖̂ − 5𝑗̂ + �̂� + 𝜆(3𝑖̂ + 2𝑗̂ + 6�̂�) and 𝑟 = 7𝑖̂ − 6�̂� − 𝜇(𝑖̂ + 2𝑗̂ + 2�̂�) [𝑐𝑜𝑠−1(19

21)]

Q2. Find the angle between the following pairs of lines:

i. 𝑥+33

=𝑦−1

5=

𝑧+3

4 and

𝑥+1

1=

𝑦−4

1=

𝑧−5

2 [𝑐𝑜𝑠−1(8√3/15)]

ii. 𝑥−22

=𝑦−1

5=

𝑧+3

−3 and

𝑥+1

−1=

𝑦−4

8=

𝑧−5

4 [𝑐𝑜𝑠−1(

26

9√38)]

Q3. i. If the lines 𝑥−1

−3=

𝑦−2

2𝑘=

𝑧−3

2 and

𝑥−1

3𝑘=

𝑦−1

1=

𝑧−6

−5 are perpendicular, then find the value of k.

[-10/7]

ii. Find the values of k so that the lines 1−𝑥

3=

𝑦−2

2𝑘=

𝑧−3

2 and

𝑥−1

3𝑘=

𝑦−1

1=

6−𝑧

7 are perpendicular to

each other. [-2]

Q4. Show that the lines x= - y = 2z and x+2 = 2y – 1 = -z +1 are perpendicular to each other.

Q5. Find the angle between the following pairs of lines:

i. 𝑥−23

=𝑦+1

−2, 𝑧 = 2 𝑎𝑛𝑑

𝑥−1

1=

𝑦+3

3=

𝑧+5

2 [𝑐𝑜𝑠−1(

3

√182)]

ii. 5−𝑥3

=𝑦+3

−2, 𝑧 = 5 𝑎𝑛𝑑

𝑥−1

1=

1−𝑦

3=

𝑧−5

2 [𝑐𝑜𝑠−1(

3

√182)]

Q6. i. Find the foot of the perpendicular drawn from the point (0,2,3) on the line 𝑥+3

5=

𝑦−1

2=

𝑧+4

3 .

[(2,3,-1)]

ii. find the foot of the perpendicular drawn from the point P(1,6,3) on the line 𝑥

1=

𝑦−1

2=

𝑧−2

3. Also

find its distance from P. [(1,3,5);√13]

iii. find the length and the foot of perpendicular drawn from the point (2,-1,5) to the line 𝑥−11

10=

𝑦+2

−4=

𝑧+8

−11. [√14, (1,2,3)]

Q7. A(0,6,-9), B(-3,-6,3) and C97,4,-1) are three points. Find the equations of the line AB. If D is the foot

of perpendicular drawn from C to the line AB, then find coordinates of the point D.

[𝑥

1=

𝑦−6

4=

𝑧+9

−4; (−1,2, −5)]

Q8. Find the image of the point P(1,6,3) in the line 𝑥

1=

𝑦−1

2=

𝑧−2

3. [(1,0,7)]

7

Q9. Find the image of the point P(2,-1,5) in the line 𝑟 = 11𝑖̂ − 2𝑗̂ − 8�̂� + 𝜆(10𝑖̂ − 4𝑗̂ − 11�̂�).

[(0,5,1)]

Q10. i. Find the equation of the line passing through the point (2,1,3) and perpendicular to the lines 𝑥−1

1=

𝑦−2

2=

𝑧−3

3 𝑎𝑛𝑑

𝑥

−3=

𝑦

2=

𝑧

5. [

𝑥−2

2=

𝑦−1

−7=

𝑧−3

4]

ii. Determine the vector equation of the line passing through the point (1,2,-4) and perpendicular

to the lines 𝑥−8

3=

𝑦+9

−16=

𝑧−10

7 𝑎𝑛𝑑

𝑥−15

3=

𝑦−29

8=

𝑧−5

−5. [𝑟 = 𝑖̂ + 2𝑗̂ − 4�̂� + 𝜆(2𝑖̂ + 3𝑗̂ + 6�̂�)]

Q11. i. Find the equation of the line through the point 𝑖̂ + 𝑗̂ − 3�̂� and perpendicular to the lines 𝑟 =

2𝑖̂ − 3𝑗̂ + 𝜆(2𝑖̂ + 𝑗̂ − 3�̂�) and 𝑟 = 3𝑖̂ − 5�̂� + 𝜇(𝑖̂ + 𝑗̂ + �̂�). [𝑟 = 𝑖̂ + 𝑗̂ − 3�̂� + 𝜆(4𝑖̂ − 5𝑗̂ + �̂�)]

ii.Find the equation of the line through the point (2,-1,3) and perpendicular to the lines 𝑟 =

−5𝑗̂ + 𝜆(𝑖̂ − 2𝑗̂ + �̂�) and 𝑟 = 7𝑖̂ − 5𝑗̂ + 9�̂� + 𝜇(𝑖̂ + 2𝑗̂ + 2�̂�).

[ 𝑟 = 2�̂� − 𝑗̂ + 3�̂� + 𝜆(−6𝑖̂ − 3𝑗̂ + 6�̂�)]

SHORTEST DISTANCE BETWEEN TWO LINES

Two straight lines which are neither parallel nor intersecting are called skew lines. SD between two skew lines

𝑟 = 𝑎1⃗⃗⃗⃗⃗ + λb1⃗⃗⃗⃗⃗ and 𝑟 = 𝑎2⃗⃗⃗⃗⃗ + λb2⃗⃗⃗⃗⃗ is

𝑆𝐷 = |(𝑎2⃗⃗⃗⃗⃗⃗ −𝑎1⃗⃗⃗⃗⃗⃗ ) ∙ (𝑏1⃗⃗ ⃗⃗ ⃗ × 𝑏2⃗⃗ ⃗⃗ ⃗)

|𝑏1⃗⃗ ⃗⃗ ⃗×𝑏2⃗⃗ ⃗⃗ ⃗||

Condition for two lines to intersect is SD = 0 ⇒ (𝑎2⃗⃗⃗⃗⃗ − 𝑎1⃗⃗⃗⃗⃗) ∙ (𝑏1⃗⃗ ⃗⃗ × 𝑏2⃗⃗⃗⃗⃗) = 0

IN CARTESIAN FORM THE CONDITIO FOR TWO LINES

𝑥 − 𝑥1𝑎1

=𝑦 − 𝑦1

𝑏1=

𝑧 − 𝑧1𝑐1

𝑥−𝑥2

𝑎2=

𝑦−𝑦2

𝑏2=

𝑧−𝑧2

𝑐2

TO INTERSECT IS |

𝑥2 − 𝑥1 𝑦2 − 𝑦1 𝑧2 − 𝑧1𝑎1 𝑏1 𝑐1𝑎2 𝑏2 𝑐2

| = 0

DISTANCE BETWEEN TWO PARALLEL LINES 𝑟 = 𝑎1⃗⃗⃗⃗⃗ + λb⃗⃗ and 𝑟 = 𝑎2⃗⃗⃗⃗⃗ + λb⃗⃗ is

SD= |�⃗⃗� ×(𝑎2⃗⃗⃗⃗⃗⃗ − 𝑎1⃗⃗⃗⃗⃗⃗ )|

|�⃗⃗�|

8

EXCERCISE 4

Q1. Find the shortest distance between each of the following lines:

i. 𝑟 = 𝑖̂ + 2𝑗̂ + �̂� + 𝜆(𝑖̂ − 𝑗̂ + �̂�) and 𝑟 = 2 𝑖̂ − 𝑗̂ − �̂� + 𝜇(2𝑖̂ + 𝑗̂ + 2�̂�) [3√2/2] ii. 𝑟 = 𝑖̂ + 2𝑗̂ + 3�̂� + 𝜆(𝑖̂ − 3�̂� + 2�̂�) and 𝑟 = 4 𝑖̂ + 5𝑗̂ + 6�̂� + 𝜇(2𝑖̂ + 3𝑗̂ + �̂�) [3/√19] iii. 𝑟 = 6𝑖̂ + 2𝑗̂ + 2�̂� + 𝜆(𝑖̂ − 2𝑗̂ + 2�̂�) and 𝑟 = −4 𝑖̂ − �̂� + 𝜇(3𝑖̂ − 2𝑗̂ − 2�̂�) [9] iv. 𝑟 = (1 + 2𝜆)𝑖̂ + (1 − 𝜆)𝑗̂ + 𝜆�̂� and 𝑟 = 2 𝑖̂ − 𝑗̂ − �̂� + 𝜇(3𝑖̂ − 5𝑗̂ + 2�̂�) [12/√59]

Q2. Find the shortest distance between the lines:

i. 𝑥−83

=𝑦+9

−16=

10−𝑧

−7 𝑎𝑛𝑑

𝑥−15

3=

58−2𝑦

−16=

𝑧−5

−5 [14]

ii. 𝑥−12

=𝑦−2

3=

𝑧−3

4 𝑎𝑛𝑑

𝑥−2

3=

𝑦−4

4=

𝑧−5

5 [1/√6;

𝑥−5/3

1=

𝑦−3

−2=

𝑧−13/3

1]

Q3. Prove that the lines 2x=y=-z and x-1 = -y-2 = -2z +1 are skew.

Q4. Find the vector equation of the line passing through the point (-1,2,1) and parallel to the line 𝑟 =

2 𝑖̂ + 3𝑗̂ − �̂� + 𝜇(𝑖̂ − 2𝑗̂ + �̂�). Also, find the distance between these lines.

𝑟 = −𝑖̂ + 2𝑗̂ + �̂� + 𝜇(𝑖̂ − 2𝑗̂ + �̂�); √83

6 ]

Q5. Find the vector equation of the line parallel to the line 𝑥−1

5=

3−𝑦

2=

𝑧+1

4 and passing through (3,0,-4).

Also, find the distance between them. 𝑟 = 3𝑖̂ − 4𝑗̂ + 𝜇(5𝑖̂ − 2𝑗̂ + 4�̂�); √974

45 ]

EQUATION OF A PLANE

NORMAL FORM

(I) Vector equation Equation of a plane perpendicular to a given direction (�̂�) and at a given

distance from the origin (p) is

𝑟 ∙ �̂� = 𝑝

(II) CARTESIAN FORM Equation of a plane whose perpendicular distance from origin is p and DC’S of the line

perpendicular to the plane is 𝑙, 𝑚, 𝑛 is given as

𝑙𝑥 + 𝑚𝑦 + 𝑛𝑧 = 𝑝

9

POINT- DIRECTION FORM

NORMAL FORM

(I) VECTOR FORM

Equation of plane passing through a given point ( �⃗�) and perpendicular to a vector �⃗⃗�

is (𝑟 − �⃗�) ∙ �⃗⃗� = 0

(II) CARTESIAN FORM

Equation of plane passing through a given point (𝑥1, 𝑦1, 𝑧1) and perpendicular to a line

having DR’S < 𝑎, 𝑏, 𝑐 > is given by

𝑎(𝑥 − 𝑥1) + 𝑏(𝑦 − 𝑦1) + 𝑐(𝑧 − 𝑧1) = 0

EXERCISE 5

Q1. Find the direction cosines of the normal to the plane 2x + 3y – z – 7 =0. []

Q2. Find a unit normal vector to the plane 2x – 2y – z – 5 = 0. [±1

3(2𝑖̂ − 2𝑗̂ − �̂�]

Q3. Find the equation of the plane passing through the point (2, -3 , 1) and perpendicular to the line

whose direction ratios are 3, -1 , 5. [3x-y+5z-14=0]

Q4. Find the equation of the plane passing through the point (4,-1,2) and perpendicular to the line 𝑥−1

2=

𝑦−2

3=

𝑧−3

4. [2x+3y+4z-13=0]

Q5. Show that the line 𝑥−2

1=

𝑦+2

−1=

𝑧−3

4 is parallel to the plane x+5y+z=7.

Q6. If the line 𝑥+1

3=

𝑦−2

4=

𝑧+6

5 is parallel to the plane 2x – 3y +kz = 0, then find the value of k.

Q7. If (3,6,11) is the image of the point (1,2,3) in the plane x + 2y + 4z + k =0, then find the value of k.

[6/5]

Q8. If (3,k,6) is the image of the point (1,3,4) in the plane x – y + z =5, then find the value of k. [1]

Q9. Write the image of the point (-2,3,5) in the plane XOY. [(-2,3,-5)]

Q10. Find the vector equation of a plane which is at a distance of 5 units from the origin and which is

normal to the vector 3𝑖̂ + 2𝑗̂ − 6�̂�. [𝑟 ∙ (3𝑖̂ + 2𝑗̂ − 6�̂�) = 35]

Q11. If the vector equation of a plane is 𝑟 ∙ (2𝑖̂ + 2𝑗̂ − �̂�) = 21, then find the length of perpendicular

from the origin to the plane. [7]

10

Q12. Find the vector and the Cartesian equation of the plane

i. That passes through the point (1,0,-2) and normal vector to the plane is 𝑖̂ + 𝑗̂ − �̂�. [𝑟 ∙ (𝑖̂ + 𝑗̂ − �̂�) = 3; 𝑥 + 𝑦 − 𝑧 = 3]

ii. That passes through the point (1,4,6) and normal to the plane is 𝑖̂ − 2𝑗̂ + �̂�. [𝑟 ∙ (3𝑖̂ − 2𝑗̂ + �̂�) = −1; 𝑥 − 2𝑦 + 𝑧 + 1 = 0]

Q13. Find the vector and Cartesian equation of the plane passing through the point (1,2,3) and

perpendicular to the line with D.R’s 2,3,-4. [𝑟 ∙ (2𝑖̂ + 3𝑗̂ − 4�̂�) = −4; 2𝑥 + 3𝑦 − 4𝑧 + 4 = 0]

Q14. Find the vector and the Cartesian equation of the plane through the point with position vector

2𝑖̂ − 𝑗̂ + �̂� and perpendicular to the vector 4𝑖̂ + 2𝑗̂ − 3�̂�.

[𝑟 ∙ (4𝑖̂ + 2𝑗̂ − 3�̂�) = 3 ; 4𝑥 + 2𝑦 − 3𝑧 = 3]

Q15. If the foot of the perpendicular drawn from the origin to the plane is (12, -4, -3), find the equation

of the plane in vector as well as Cartesian form. [𝑟 ∙ (12𝑖̂ − 4𝑗̂ − 3�̂�) = 169; 12𝑥 − 4𝑦 − 3𝑧 = 169]

Q16. Find the equation in Cartesian form of the plane passing through the point (2, -3,1) and

perpendicular to the line joining the points (3,4,-1) and (2,-1,5). [x+5y-6z+19=0]

Q17. Find the equation of the plane which bisects the line joining the points (-1,2,3) and (3,-5,6) at right

angles. [4x-7y+3z-28=0]

Q18. Change the equation of the plane 2x – 3y + 6z + 14 = 0 to normal form. Hence, find the length of

perpendicular from origin to the plane. [−2

7𝑥 +

3

7𝑦 −

6

7𝑧 = 2; 2]

Q19. Find the vector equation of the line passing through the point 91,-1,2) and perpendicular to the

plane 2x –y + 3z = 5. [𝑟 = 𝑖̂ − 𝑗̂ + 2�̂� + 𝜆(2𝑖̂ − 𝑗̂ + 3�̂�)]

Q20. Find the equation of the line passing through the point with position vector 𝑖̂ − 𝑗̂ + 2 �̂� and

perpendicular to the plane 𝑟 ∙ (2𝑖̂ − 𝑗̂ + 3 �̂�) = 5. [𝑟 = 𝑖̂ − 𝑗̂ + 2�̂� + 𝜆(2𝑖̂ − 𝑗̂ + 3�̂�)]

Q21. Find the point where the line 𝑥−1

2=

𝑦−2

−3=

𝑧+3

4 meets the plane 2x + 4y – z =1. [(3,-1,1)]

Q22. Find the coordinates of the point where the line through (3, -4 , -5) and (2, -3 ,1) crosses the plane

2x + y + z =7. [1,-2,7)]

Q23. Show that the distance of the point of intersection of the line 𝑥−2

3=

𝑦+1

4=

𝑧−2

12 and the plane x – y

+ z = 5 from the point whose position vector is −𝑖̂ − 5𝑗̂ − 10�̂� is 13.

Q24. Find the equation of the line passing through the point P(4,6,2) and the point of intersection of the

line 𝑥−1

3=

𝑦

2=

𝑧+1

7 and the plane x + y – z =8. [

𝑥−4

1=

𝑦−6

1=

𝑧−2

2]

11

Q25. Find the vector equation of a line passing through the point with position vector 2𝑖̂ − 3𝑗̂ − 5�̂� and

perpendicular to the plane 𝑟 ∙ (6𝑖̂ − 3𝑗̂ + 5�̂�) + 2 = 0. Also find the point of intersection of this line

and the plane. [𝑟 = 2𝑖̂ − 3𝑗̂ − 5�̂� + 𝜆(6𝑖̂ − 3𝑗̂ + 5�̂�); (76

35, −

108

35, −

34

7)]]

Q26. Find the distance of the point (3,4,5) from the plane x+y+z =2 measured parallel to the line

2x = y =z. [6]

Q27. Find the length and the foot of perpendicular drawn from the point (2,3,7) to the plane

3x – y – z = 7. [√11, (5,2,6)]

Q28. Find the length and the foot of perpendicular drawn from (1,1,2) to the plane

𝑟 ∙ (2𝑖̂ − 2 𝑗̂ + 4 �̂�) + 5 = 0. [13

12√6; (−

1

12,

25

12, −

1

6)]

Q29. From the point P(1,2,4), a perpendicular is drawn on the plane 2x + y – 2z +3 =0. Find the equation,

the length and the coordinates of foot of perpendicular. [𝑥−1

2=

𝑦−2

1=

𝑧−4

−2;

1

3; (

11

9,

19

9,

34

9)]

Q30. Find the coordinates of the image of the point (1,3,4) in the plane 2x – y + z + 3 =0. [(-3,5,2)]

Q31. Prove that the image of the point (3,-2,1) in the plane 3x – y + 4z =2 lies on the plane x+y+z+4 = 0.

EQUATION OF PLANE PASSING THROUGH TWO GIVEN POINTS AND PARALLEL TO A

GIVEN LINE

(I) VECTOR FORM Given points A(𝑝. 𝑣. 𝑟1⃗⃗⃗ ⃗) and B( 𝑝. 𝑣. 𝑟2⃗⃗⃗⃗ ) and parallel to the line 𝑟 = �⃗� + 𝜆 �⃗⃗� is given by

(𝑟 − 𝑟1⃗⃗⃗ ⃗ ) ∙ ( ( 𝑟2⃗⃗⃗⃗ − 𝑟1⃗⃗⃗ ⃗ ) × �⃗⃗� ) = 0

(II) CARTESIAN FORM Given points A (𝑥1, 𝑦1, 𝑧1) 𝑎𝑛𝑑 𝐵 ( 𝑥2, 𝑦2, 𝑧2) and parallel to the line

𝑥−𝑥′

𝑎=

𝑦−𝑦′

𝑏=

𝑧−𝑧′

𝑐 is

given by |𝑥 − 𝑥1 𝑦 − 𝑦1 𝑧 − 𝑧1𝑥2 − 𝑥1 𝑦2 − 𝑦1 𝑧2 − 𝑧1

𝑎 𝑏 𝑐| = 0

EQUATION OF PLANE THROUGH A GIVEN POINT AND PARALLEL TO TWO

(NON – PARALLEL) LINES

(I) VECTOR FORM Given point A ( p.v. �⃗�) and the lines 𝑟 = �⃗⃗� + 𝜆 𝑏1⃗⃗ ⃗⃗ 𝑎𝑛𝑑 𝑟 = 𝑐 + 𝜇 𝑏2⃗⃗⃗⃗⃗ ,

then equation of plane is given by (𝑟 − �⃗�) ∙ (𝑏1⃗⃗ ⃗⃗ × 𝑏2⃗⃗⃗⃗⃗ ) = 0

(II) CARTESIAN FORM Given point A ( 𝑥1, 𝑦1, 𝑧1) 𝑎𝑛𝑑 𝑙𝑖𝑛𝑒𝑠

𝑥 − 𝑥′

𝑎1=

𝑦 − 𝑦′

𝑏1=

𝑧 − 𝑧′

𝑐1

12

𝑥 − 𝑥′′

𝑎2=

𝑦 − 𝑦′′

𝑏2=

𝑧 − 𝑧′′

𝑐2

is given by |

𝑥 − 𝑥1 𝑦 − 𝑦1 𝑧 − 𝑧1𝑎1 𝑏1 𝑐1𝑎2 𝑏2 𝑐2

| = 0

CONDITION OF COPLANARITY OF TWO LINES

𝑥 − 𝑥1𝑎1

=𝑦 − 𝑦1

𝑏1=

𝑧 − 𝑧1𝑐1

𝑥−𝑥2

𝑎2=

𝑦−𝑦2

𝑏2=

𝑧−𝑧2

𝑐2 is given by

|

𝑥2 − 𝑥1 𝑦2 − 𝑦1 𝑧2 − 𝑧1𝑎1 𝑏1 𝑐1𝑎2 𝑏2 𝑐2

| = 0

EQUATION OF PLANE CONTAINING TWO LINES

𝑥 − 𝑥1𝑎1

=𝑦 − 𝑦1

𝑏1=

𝑧 − 𝑧1𝑐1

𝑥−𝑥2

𝑎2=

𝑦−𝑦2

𝑏2=

𝑧−𝑧2

𝑐2 is given by

|

𝑥 − 𝑥1 𝑦 − 𝑦1 𝑧 − 𝑧1𝑎1 𝑏1 𝑐1𝑎2 𝑏2 𝑐2

| = 0

EQUATION OF PLANE PASSING THROUGH THREE POINTS

𝐴(𝑥1, 𝑦1, 𝑧1), 𝐵(𝑥2, 𝑦2 , 𝑧2)𝑎𝑛𝑑 𝐶(𝑥3, 𝑦3, 𝑧3) is given by

|

𝑥 − 𝑥1 𝑦 − 𝑦1 𝑧 − 𝑧1𝑥2 − 𝑥1 𝑦2 − 𝑦1 𝑧2 − 𝑧1𝑥3 − 𝑥1 𝑦3 − 𝑦1 𝑧3 − 𝑧1

| = 0

13

EQUATION OF PLANE IN INTERCEPT FORM

Equation of the plane when

𝑥 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 → 𝑎, 𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 → 𝑏 𝑎𝑛𝑑 𝑧 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 → 𝑐 is given by

𝑥

𝑎+

𝑦

𝑏+

𝑧

𝑐= 1

EQUATION OF A PLANE PASSING THROUGH THE INTERSECTION OF TWO

PLANES 𝑷𝟏 𝑨𝑵𝑫 𝑷𝟐 IS GIVEN BY

𝑃1 + 𝜆𝑃2 = 0

EXCERCISE 6

Q1. Write the intercept cut off by the plane 2x + y – z = 5 on (i) x- axis (ii) y-axis (iii) z-axis. [ 5/2, 5, -5]

Q2. Write the equation of the plane with intercepts 2, -3 and 5 on the co-ordinate planes respectively.

[15x-10y+6z=30]

Q3. Find the equation of the plane passing through (2,3,-4) and (1,-1,3) and parallel to x-axis.

[7y+4z-5=0]

Q4. Find the equation of the plane containing the line 𝑥−3

2=

𝑦+2

9=

𝑧−4

−1 and the point (-6,3,2).

[x-y-7z+23=0]

Q5. Find the foot of the perpendicular drawn from the point P(1,2,3) on the line 𝑥−6

3=

𝑦−7

2=

𝑧−7

−2. Also

obtain the equation of the plane containing the line and the point (1,2,3). [(3,5,9); 18x – 22y + 5z +11=0]

Q6. Find the Cartesian equation of the plane passing through the points A(0,0,0) and B(3,-1,2) and

parallel to the line 𝑥−4

1=

𝑦+3

−4=

𝑧+1

7. [x-19y-11z=0]

Q7. Find the equation of the plane passing through the points (3,4,1) and (0,1,0) and parallel to the line 𝑥+3

2=

𝑦−3

7=

𝑧−2

5. [8x-13y+15z+13=0]

Q8. Find the vector equation of the plane passing through the points (0, - 4, 1) and (2,3,-2) and parallel

to the line 𝑟 = 5𝑖̂ − 2𝑗̂ + �̂� + 𝜆(2𝑖̂ − �̂�). [ 𝑟 ∙ (7𝑖̂ + 4𝑗̂ + 14�̂�) + 2 = 0]

Q9. Find the vector equation of the plane passing through the point 𝑖̂ + 2𝑗̂ + 3�̂� and parallel to the lines

whose direction ratios are 1, -1 , -2 and -1, 0 ,2. [𝑟 ∙ (2𝑖̂ + �̂�) = 5]

14

Q10. Find the equation of the plane passing through the point (1,2,- 4) and parallel to the lines 𝑥−1

2=

𝑦−2

3=

𝑧+1

6 and

𝑥−1

1=

𝑦+3

1=

𝑧

−1. [9x-8y+z+11=0]

Q11. Find the equation of the plane containing the line 𝑥−4

2=

𝑦−2

3=

𝑧−3

4 and parallel to the line

𝑥

1=

𝑦−1

3=

𝑧−2

3. [3x+2y-3z-7=0]

Q12. Find the vector equation of the plane which contains the two parallel lines 𝑥−4

1=

𝑦−3

−4=

𝑧−2

5 𝑎𝑛𝑑

𝑥−3

1=

𝑦+2

−4=

𝑧

5. [𝑟 ∙ (11𝑖̂ − 𝑗̂ − 3�̂�) = 35]

Q13. Show that the lines 𝑥+3

−3=

𝑦−1

1=

𝑧−5

5 and

𝑥+1

−1=

𝑦−2

2=

𝑧−5

5 are coplanar. Also find the equation of

the plane containing the lines. [x-2y+z=0]

Q14. Show that the lines 𝑥+1

−3=

𝑦−3

2=

𝑧+2

1 𝑎𝑛𝑑

𝑥

1=

𝑦−7

−3=

𝑧+7

2 intersect. Find the point of intersection

and the equation of the plane containing them. [(2,1,-3); x+y+z=0]

Q15. Find the equation of the plane containing the lines 𝑥−4

1=

𝑦−3

4=

𝑧−2

5 𝑎𝑛𝑑

𝑥−3

1=

𝑦−2

−4=

𝑧+3

5.

[5x-z-18=0]

Q16. Find the vector and Cartesian equation of the plane passing through the points R(2,5,-3), S(-2,-3,5)

and T(5,3,-3). [2x+3y+4z=7;𝑟 ∙ (2𝑖̂ + 3𝑗̂ + 4�̂�) = 7]

Q17. Find the vector equation of the plane passing through the points A(2,2,-1), B(3,4,2) and C(7,0,6).

Also find the Cartesian equation of the plane. [𝑟 ∙ (5𝑖̂ + 2𝑗̂ − 3�̂�) = 17]

Q18. Find the equation of the plane passing through the point (2,4,6) and making equal intercepts on

axes. [𝑥 + 𝑦 + 𝑧 = 12]

Q19. A plane meets the coordinate axes at A, B and C respectively such that centroid of triangle ABC is

(1, -2 ,3). Find the equation of the plane. [6x-3y+2z-18=0]

Q20. Find the equation of the plane passing through the intersection of the planes x+2y+3z +4=0 and

x- y +z+3=0 and passing through the origin. [x-10y-5z=0]

Q21. Find the equation of the plane passing through the intersection of the planes 3x-y +2z-4=0 and

x+y+z-2=0 and the point (2,2,1). [7x-5y+4z-8=0]

Q22. Find the equation of the plane passing through the intersection of the planes

𝑟 ∙ (2𝑖̂ − 7𝑗̂ + 4�̂�) = 3 and 𝑟 ∙ (3𝑖̂ − 5𝑗̂ + 4�̂�) + 11 = 0 and passing through the point (-2,1,3).

[𝑟 ∙ (15𝑖̂ − 47𝑗̂ + 28�̂�) = 7]

15

ANGLE BETWEEN TWO PLANES

(I) VECTOR FORM

�⃗⃗⃗� ∙ 𝑛1⃗⃗⃗⃗⃗ = 𝑑1 and 𝑟 ∙ 𝑛2⃗⃗⃗⃗⃗ = 𝑑2 is given by

cos 𝜃 = 𝑛1⃗⃗⃗⃗⃗⃗ ∙𝑛2⃗⃗⃗⃗⃗⃗

|𝑛1⃗⃗⃗⃗⃗⃗ ||𝑛2⃗⃗⃗⃗⃗⃗ |

(II) CARTESIAN FORM

𝑎1𝑥 + 𝑏1𝑦 + 𝑐1𝑧 + 𝑑1 = 0 𝑎𝑛𝑑 𝑎2𝑥 + 𝑏2𝑦 + 𝑐2𝑧 + 𝑑2 = 0 is given by

𝑐𝑜𝑠𝜃 = 𝑎1𝑎2 + 𝑏1𝑏2 + 𝑐1𝑐2

√𝑎12 + 𝑏12 + 𝑐12√𝑎22 + 𝑏2

2 + 𝑐22

ANGLE BETWEEN A LINE AND A PLANE (I) VECTOR FORM

Angle between the plane 𝑟 ∙ �⃗⃗� = 𝑑 𝑎𝑛𝑑 𝑡ℎ𝑒 𝑙𝑖𝑛𝑒 𝑟 = �⃗� + 𝜆�⃗⃗� is given

by 𝑠𝑖𝑛𝜃 = �⃗⃗�∙�⃗⃗�

|�⃗⃗�||�⃗⃗�|

(II) CARTESIAN FORM Angle between the plane

𝐴𝑥 + 𝐵𝑦 + 𝐶𝑧 + 𝐷 = 0 𝑎𝑛𝑑 𝑡ℎ𝑒 𝑙𝑖𝑛𝑒 𝑥−𝑥1

𝑎=

𝑦−𝑦1

𝑏=

𝑧−𝑧1

𝑐 is given by

𝑠𝑖𝑛𝜃 = 𝑎𝐴 + 𝑏𝐵 + 𝑐𝐶

√𝑎2 + 𝑏2 + 𝑐2√𝐴2 + 𝐵2 + 𝐶2

EXERCISE 7

Q1. Find the angle between the planes :

i. 𝑟 ∙ (𝑖̂ + 𝑗̂) = 1𝑎𝑛𝑑 𝑟 ∙ (𝑖̂ + �̂�) = 3 [60°] ii. 𝑟 ∙ (2𝑖̂ − 𝑗̂ + 2�̂�) = 6 𝑎𝑛𝑑 𝑟 ∙ (3𝑖̂ + 6𝑗̂ − 2�̂�) = 9 [𝑐𝑜𝑠−1( 4

21)]

iii. 𝑟 ∙ (3𝑖̂ − 4𝑗̂ + 5�̂�) = 0 𝑎𝑛𝑑 𝑟 ∙ (2𝑖̂ − 𝑗̂ − 2�̂�) = 7 [90°]

iv. 3x - 6y+2z =7 and 2x + 2y -2z =5 [𝑐𝑜𝑠−1(5√321

)

v. 2x – y +z =6 and x+ y +2z =7 [60°]

16

Q2. Show that the following pairs of planes are at right angles to each other:

i. 3x – 5y + 3z – 1 =0 and 2x + 3y + 3z =7 ii. 𝑟 ∙ (2𝑖̂ + 6𝑗̂ + 6�̂�) = 13 𝑎𝑛𝑑 𝑟 ∙ (3𝑖̂ + 4𝑗̂ − 5�̂�) + 7 = 0

Q3. Find the value of p for which the following planes are perpendicular to each other:

i. 𝑟 ∙ (𝑝𝑖̂ + 2𝑗̂ + 3�̂�) = 5 𝑎𝑛𝑑 𝑟 ∙ (𝑖̂ + 2𝑗̂ − 7�̂�) + 11 =0 [17] ii. 𝑥 − 4𝑦 + 𝑝𝑧 + 3 = 0 𝑎𝑛𝑑 2𝑥 + 2𝑦 + 3𝑧 = 5 [2] iii. 3𝑥 − 6𝑦 − 2𝑧 = 7 𝑎𝑛𝑑 2𝑥 + 𝑦 − 𝑝𝑧 = 5 [0]

Q4. Write the equation of the plane passing through origin and parallel to the plane

5x – 3y + 7z + 13 = 0. [5x-3y+7z=0]

Q5. Find the equation of the plane through the point (1,4,-2) and parallel to the plane – 2x +y – 3z =7.

[2x-y+3z+8=0]

Q6. Find the equation of the plane through the point (3𝑖̂ + 4𝑗̂ − �̂�) and parallel to the plane

𝑟 ∙ (2𝑖̂ − 3𝑗̂ + 5�̂�) + 5 = 0. [𝑟 ∙ (2𝑖 − 3𝑗 + 5𝑘) + 11 = 0]

Q7. Find the angle between the line 𝑥−2

3=

𝑦+1

−1=

𝑧−3

2 and the plane 3x + 4y + z + 5 =0.[𝑠𝑖𝑛−1(

7

2√91)]

Q8. Prove that the plane 2𝑥 + 3𝑦 − 4𝑧 = 11 is perpendicular to each of the planes x + 2y + 2z – 7 =0

and 5x + 6y + 7z = 23.

Q9. Find the equation of the plane through the points (2,1,-1) and (-1 , 3 4) and perpendicular to the

plane x – 2y + 4z =10. [18x+17y+4z-49=0]

Q10. Find the equation of the plane through the points (2,2,1) and (9,3,6) and perpendicular to the

plane 2x + 6y + 6z = 1. [3x+4y-5z-9=0]

Q11. Find the equation of the plane through the point (1,2,3) and perpendicular to each of the plane

x + y + 2z = 3 and 3x + 2y + z = 4. [3x-5y+z+4=0]

Q12. Find the equation of the plane through the point ( -1 , -1 ,2) and perpendicular to the planes

2x + 3y – 3z =2 and 5x – 4y + z =6. [9x+17y+23z=20]

Q13. Find the equation of the plane passing through the point ( -1 , 3 , 2) and perpendicular to each of

the planes x + 2y + 3z = 5 and 3x + 3y + z =0. [7x-8y+3z+25=0]

Q14. Find the vector equation of the plane passing through the point (5𝑖̂ + 2𝑗̂ − 3�̂�) and perpendicular

to the line of intersection of the planes 𝑟 ∙ (2𝑖̂ − 𝑗̂ + 2�̂�) = 0 and 𝑟 ∙ (𝑖̂ + 3𝑗̂ − 5�̂�) + 7 = 0.

[𝑟 ∙ (2𝑖̂ − 4𝑗̂ + 3�̂�) = 8

Q15. Find the equation of the plane containing the line 𝑥−1

2=

𝑦+1

1=

𝑧−3

4 and perpendicular to the plane

x + 2y – z =12. [3x-2y-z-2=0]

17

Q16. Find the equation of the plane through the line of intersection of the planes x + y + z = 1

and 2x + 3y + 4z = 5 and perpendicular to the plane x – y + z =0. [x-z+2=0]

Q17. Find the equation of the plane which is perpendicular to the plane 5x + 3y + 6z +8 =0 and which

contains the line of intersection of the planes x + 2y + 3z – 4 =0 and 2x + y – z =0.

[51x+15y-50z-28=0]

Q18. Find the equation of the plan e passing through the line of intersection of the planes 2x + 3y – z + 1

=0 and x + y – 2z + 3 =0 and perpendicular to the planes 3x – y – 2z – 4 =0. Also find the inclination of

this plane with the xy- plane. [7x+13y+4z-9=0; 𝑐𝑜𝑠−1(4

√234)]

Q19. Find the equation of the plane passing through the line of intersection of the planes

2x + 3y – 4z + 5 = 0 and x – 5y + 7z + 2 =0 and makes intercepts on y-axis equal to 3.

[40x – 31y +46z +93 = 0]

Q20. Find the equation of the plane through the line of intersection of the planes

3x – 4y + 5z = 10 and 2x + 2y – 3z = 4 and parallel to the line x = 2y = 3z.

[x-20y+27z = 14]

DISTANCE OF A POINT P(𝒙𝟏, 𝒚𝟏, 𝒛𝟏) FROM THE PLANE AX +By +Cz +D =0 is given by

Dist = |𝑨𝒙𝟏+𝑩𝒚𝟏+𝑪𝒛𝟏+𝑫

√𝑨𝟐+𝑩𝟐+𝑪𝟐|

DISTANCE BETWEEN TWO PARALLEL PLANES

Ax + By + Cz + 𝑫𝟏 = 𝟎 AND Ax + By + Cz + 𝑫𝟐 = 𝟎

|𝑫𝟐−𝑫𝟏

√𝑨𝟐+𝑩𝟐+𝑪𝟐|

18

EXERCISE 8

Q1. Find the perpendicular distance

i. From the point (0,0,0) to the plane 3x – 4y +12z =3. [3/13] ii. From the point (-6,0,0) to the plane 2x – 3y + 6z -2 =0. [2] iii. From the point (2,3,-5) to the plane x + 2y – 2z = 9. [3] iv. From the point (3, -2, 1) to the plane 2x – y + 2z +3 =0. [13/3]

Q2. Write the distance of the point (3,4,5) from the plane 𝑟 ∙ (2𝑖̂ − 5𝑗̂ + 3�̂�) = 13 [12/√38]

Q3. Write the distance from the point (2,3,- 5 ) to the XY- plane. [5]

Q4. If the distance from the point (- 6, 0,0) to the plane 2x – 3y +z – k =0 is √14 units, find the value of k.

[2,-26]

Q5. Find the distance between the planes 2x + 3y + 4z = 4 and 4x + 6y + 8z =12. [2/√29]

Q6. Show that the points (𝑖̂ − 𝑗̂ + 3�̂�) and (3𝑖̂ + 3𝑗̂ + 3�̂�) are equidistant from the plane 𝑟 ∙

(5𝑖̂ + 2𝑗̂ − 7�̂�) + 9 = 0.

Q7. Find the equation of the plane passing through the intersection of the planes 4x – y +z = 10

and x + y – z = 4 and parallel to the line with direction ratios 2,1,1. Also find the perpendicular distance

of the point (1,1,1) from this plane. [5y-5z-6=0; 3√2/5]

Q8. Find the equation of the plane containing the lines 𝑟 = 𝑖̂ + 𝑗̂ + 𝜆(𝑖̂ + 2𝑗̂ − �̂�) and

𝑟 = 𝑖̂ + 𝑗̂ + 𝜆(−𝑖̂ + 𝑗̂ − 2�̂�). Find the distance of this plane from the origin and also from the point

(1,1,1). [x-y-z=0; 0 and 1/√3]

Q9. Find the equation of the planes parallel to the plane x – 2y + 2z =3 and at a unit distance from the

point (1,1,1). [x-2y+2z+2=0; x-2y+2z-4=0]

Q10. Find the equation of the plane parallel to the plane 3x – 6y + 2z = 12 and 6 units away from it.

[3x-6y+2z+30=0; 3x-6y+2z-54=0]

Q11. Find the equation of the plane mid-parallel to the planes 2x – 2y + z + 3 =0 and 2x – 2y + z + 9 =0.

[ 2x-2y+z+6=0]

Q12. Show that the plane 𝑟 ∙ (𝑖̂ + 2𝑗̂ − �̂�) = 1 and the line 𝑟 = −𝑖̂ + 𝑗̂ + �̂� + 𝜆(2𝑖̂ + 𝑗̂ + 4�̂�)

are parallel. Also find the distance between them. [1/√6]