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NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry Ex 11.1

Ex 11.1 Class 12 Maths Question 1.
If a line makes angles 90°, 135°, 45° with the and z axes respectively, find its direction cosines.
Solution:
Direction angles are 90°, 135°, 45°
Direction cosines are
NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry 1

Ex 11.1 Class 12 Maths Question 2.
Find the direction cosines of a line which makes equal angles with coordinate axes.
Solution:
Let direction angle be α each
∴ Direction cosines are cos α, cos α, cos α
But l² + m² + n² = 1
∴cos² a + cos² a + cos² a = 1
NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry 2

Ex 11.1 Class 12 Maths Question 3.
If a line has the direction ratios – 18,12, -4 then what are its direction cosines?
Solution:
Now given direction ratios of a line are -8,12,-4
∴ a = -18,b = 12,c = -4
Direction cosines are
tiwari academy class 12 maths Chapter 11 Three Dimensional Geometry 3
NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry 3.1

Ex 11.1 Class 12 Maths Question 4.
Show that the points (2,3,4) (-1,-2,1), (5,8,7) are collinear.
Solution:
Let the points be A(2,3,4), B (-1, -2,1), C (5,8,7).
Let direction ratios of AB be
NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry 4

Ex 11.1 Class 12 Maths Question 5.
Find the direction cosines of the sides of the triangle whose vertices are (3,5, -4), (-1,1,2) and (-5,-5,-2).
Solution:
The vertices of triangle ABC are A (3, 5, -4), B (-1,1,2), C (-5, -5, -2)
(i) Direction ratios of AB are (-4,-4,6)
Direction cosines are
tiwari academy class 12 maths Chapter 11 Three Dimensional Geometry 5
NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry 5.1

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