If \(θ1,θ2, θ3\) are the angles made by a line with the positive directions of the \(x,y,z\) axes, then the value of \(\cos 2\theta_1 + \cos 2\theta_2 + \cos 2\theta_3\) is
If \(θ1,θ2, θ3\) are the angles made by a line with the positive directions of the \(x,y,z\) axes, then the value of \(\cos 2\theta_1 + \cos 2\theta_2 + \cos 2\theta_3\) is
\(1\)
\(-1\)
\(2\)
\(-2\)
\(0\)
The Correct Option is B
Solution and Explanation
Given that \(\theta_1, \theta_2,\) and \(\theta_3\) are the angles made by a line with the positive directions of the \(x, y,\) and \(z\) axes, respectively, we know from the property of direction cosines that:
\[ \cos^2 \theta_1 + \cos^2 \theta_2 + \cos^2 \theta_3 = 1 \]
We are asked to find the value of \(\cos 2\theta_1 + \cos 2\theta_2 + \cos 2\theta_3\). Using the double-angle identity for cosine, \(\cos 2\theta = 2\cos^2 \theta - 1\), we can rewrite the expression as:
\[ \cos 2\theta_1 + \cos 2\theta_2 + \cos 2\theta_3 = 2\cos^2 \theta_1 - 1 + 2\cos^2 \theta_2 - 1 + 2\cos^2 \theta_3 - 1 \]
Simplify the expression:
\[ = 2(\cos^2 \theta_1 + \cos^2 \theta_2 + \cos^2 \theta_3) - 3 \]
Substitute the direction cosines property:
\[ = 2(1) - 3 = -1 \]
Thus, the value of \(\cos 2\theta_1 + \cos 2\theta_2 + \cos 2\theta_3\) is \(-1\).
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Three Dimensional Geometry
Mathematically, Geometry is one of the most important topics. The concepts of Geometry are derived w.r.t. the planes. So, Geometry is divided into three major categories based on its dimensions which are one-dimensional geometry, two-dimensional geometry, and three-dimensional geometry.
Direction Cosines and Direction Ratios of Line:
Consider a line L that is passing through the three-dimensional plane. Now, x,y and z are the axes of the plane and α,β, and γ are the three angles the line makes with these axes. These are commonly known as the direction angles of the plane. So, appropriately, we can say that cosα, cosβ, and cosγ are the direction cosines of the given line L.
