Mathematics Part-II

Book: Mathematics Part-II

Chapter: 11. Three Dimensional Geometry

Subject: Maths - Class 12th

Q. No. 11 of Exercise 11.2

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11

. Find the angle between the following pair of lines:

We know that

If and are the equations of two lines, then the acute angle between the two lines is given by


cos θ = | l1l2 + m1m2 + n1n2 | ……(i)


(i) and


Here, a1 = 2, b1 = 5, c1 = -3 and a2 = -1, b2 = 8, c2 = 4


Now, ……(ii)


Here,


And


So, from (ii), we have


⇒


And


∴ From (i), we have


⇒


⇒


(ii) and


Here, a1 = 2, b1 = 2, c1 = 1 and a2 = 4, b2 = 1, c2 = 8


Here,


And


So, from (ii), we have


⇒


And


∴ From (i), we have


⇒


⇒


11

Chapter Exercises

Exercise 11.1
Exercise 11.2
Exercise 11.3
Miscellaneous Exercise

More Exercise Questions

1

Show that the three lines with direction cosines are mutually perpendicular.
2

Show that the line through the points (1, –1, 2), (3, 4, –2) is perpendicular to the line through the points (0, 3, 2) and (3, 5, 6).
3

Show that the line through the points (4, 7, 8), (2, 3, 4) is parallel to the line through the points (–1, –2, 1), (1, 2, 5).
4

Find the equation of the line which passes through the point (1, 2, 3) and is parallel to the vector

5

Find the equation of the line in vector and in cartesian form that passes through the point with position vector and is in the direction

6

Find the cartesian equation of the line which passes through the point (–2, 4, –5) and parallel to the line given by

7

The Cartesian equation of a line is . Write its vector form.
1

Find the vector and the cartesian equations of the lines that passes through the origin and (5, –2, 3).
9

. Find the vector and the cartesian equations of the line that passes through the points (3, –2, –5), (3, –2, 6).
10

Find the angle between the following pairs of lines:
11

. Find the angle between the following pair of lines:
12

Find the values of p so that the lines are at right angles.
13

Show that the lines are perpendicular to each other.
14

Find the shortest distance between the lines
15

Find the shortest distance between the lines
16

Find the shortest distance between the lines whose vector equations are
17

Find the shortest distance between the lines whose vector equations are