The acute angle which the line with direction cosines 1/√ 3, 1/√ 6, n makes with positive direction of z-axis is
An acute angle is an angle that measures less than 90 degrees but greater than 0 degrees. It is one of the basic types of angles in geometry. Acute angles are commonly found in triangles and other geometric figures and are important in various applications like trigonometry and physics.
Class 12 Maths Chapter 11 Three Dimensional Geometry is important for CBSE Exam 2024-25. It includes topics like direction cosines and direction ratios of a line equations of a line in different forms shortest distance between two lines equations of a plane angles between planes and distance of a point from a plane.
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The direction cosines of the line are given as:
l = 1/√3, m = 1/√6, n = n
The angle θ that the line makes with the positive z-axis is related to the direction cosine n by:
cos θ = n
To find the acute angle, we need to calculate the value of n using the condition that the sum of the squares of the direction cosines equals 1:
l² + m² + n² = 1
Substituting the values of l and m:
(1/√3)² + (1/√6)² + n² = 1
1/3 + 1/6 + n² = 1
1/2 + n² = 1
n² = 1/2
n = 1/√2
Thus, cos θ = 1/√2, which gives: θ = π/4
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