RB 12 Maths

RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Ex 14.2

RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Ex 14.2

Rajasthan Board RBSE Class 12 Maths Chapter 14 Three Dimensional Geometry Ex 14.2

Question 1.
Find the equation of the line which passes through the points (5, 7,9) and is parallel to the following axis :
(i) A-axis
(ii) F-axis
(iii) Z-axis
Solution:
Let position vector of point (5, 7, 9)

RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Ex 14.2
(i) The line parallel to X-axis passes through point (1,0,0) whose position vector
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Ex 14.2
∴ Vector equation of required line.
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Ex 14.2
Cartesian equation of line :
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Ex 14.2

(ii) Let the line parallel to F-axis passes through point (0, 1,0).
∴ Position vector of point (0, 1, 0).
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Ex 14.2
∴ required vector equation of line
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Ex 14.2
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Ex 14.2

(iii) The line parallel to Z-axis passes through (0, 0, 1).
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Ex 14.2
∴ Required vector equation of line
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Ex 14.2

Question 2.
Find the equation of the line in vector and in cartesian form that passes through the point with vector
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Ex 14.2
and
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Ex 14.2
is parallel to the vector
Solution:
Position vector of the given point is
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Ex 14.2
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Ex 14.2

Question 3.
Find the equation of the line parallel to the vector
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Ex 14.2
and passes through the point (5,-2,4).
Solution :
Line passes through (5, -2, 4).
∴ Position vector of (5, – 2, 4)
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Ex 14.2

Question 4.
Find the equation of the line passes through the point (2,-1,1) and is parallel to the line x3/2 = y+1/ 7 = z2/ 3
Solution:
The equation of line passing through point (2,-1, 1) and parallel to the given line x3/2 = y+1/ 7 = z2/ 3 because dc’s of parallel lines are same.
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Ex 14.2
For vector equation of line passing through (2, – 1, 1) and parallel to m,
The position vector of (2, -1, 1).
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Ex 14.2

Question 5.
If cartesian equation of line is x5/3 = y+4/7 = z6/2, then find vector equation of line.
Solution:
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Ex 14.2

Question 6.
Find the equation of a line in cartesian form which passes through (1, 2, 3) and is parallel to x2/1 = y+3/7 = 2z6/3
Solution:
Let a, b, c are dc’s of line passing through (x1,y1,z1) then equation of the line
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Ex 14.2
Here line passes through point (1, 2, 3) and parallel to line
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Ex 14.2

Question 7.
Coordinates of three vertices of parallelogram ABCD are A(4, 5, 10), B(2, 3, 4) and C(1, 2, – 1). Find vector and cartesian equation of AB and AC. Also Find coordinates of D.
Solution:
Let O is origin.
∴ Position vector of points A, B and C are respectively.
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Ex 14.2
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Ex 14.2
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Ex 14.2
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Ex 14.2
(iii) For coordinates of point D.
Let (x1, y1, z1,) are the coordinates of point D.
∵ ABCD is a parallelogram whose diagonals AC and BD bisect each other at point D. Therefore mid points of AC and BD will coincide. So the coordinate of mid point of AC.
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Ex 14.2

Question 8.
Cartesian equation of a line is 3x + 1 = 6y – 2 = 1 -z. Find the point from where it passes and also find its direction ratios and vector equation.
Solution:
The equation of given line
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Ex 14.2
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Ex 14.2

Question 9.
Find an equation passes through (1,2,3) and is parallel to the vector
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Ex 14.2
Solution:
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Ex 14.2

Question 10.
Find the equation of line in vector and cartesian form that passes through the point with position vector
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Ex 14.2
and is in the direction
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Ex 14.2
Solution:
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Ex 14.2
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Ex 14.2

Question 11.
Find the cartesian equation of a line passes through the point (- 2, 4, -5) and is parallel to
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Ex 14.2
Solution:
Let line passes through point (x1,y1,z1) whose dc’s are a,b,c then equation of line is
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Ex 14.2

Question 12.
The cartesian equation of a line is
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Ex 14.2
Write its vector form.
Solution:
The cartesian equation of line is
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Ex 14.2

Question 13.
Find the equation of line in vector and cartesian form that passes through origin and (5, -2, 3).
Solution:
(i) Position vector of origin O(0,0,0) is  =  and position vector equation of (5, – 2, 3) is

RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Ex 14.2
(ii) Line passing through origin O(0, 0, 0) whose dc’s are 5,-2, 3.
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Ex 14.2

Question 14.
Find the equation of line in vector and cartesian form that passes through the points (3, -2,-5) and (3,-2,6).
Solution:
Let line passes through point (3,-2,-5) and (3,-2,6).
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Ex 14.2
(ii) Line passes through A(3, -2,-5) and 5(3, – 2, 6). Cartesian equation of line AB
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Ex 14.2

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