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 \(cos^{2}θ_1 +cos^{2}θ_2+cos^{2}θ_3\) is
\(-1\)
\(1\)
\(2\)
\(-2\)
\(0\)
The Correct Option is B
Solution and Explanation
Given that
The line makes angles with the\( x, y,z\) axes, the direction cosines of the line are given by \(cos(θ_1), cos(θ_2),cos(θ_3)\) for the \(x, y,z\) directions, respectively.
Then we know that,
The direction cosines satisfy the relation:
\(cos^{2}θ_1 +cos^{2}θ_2+cos^{2}θ_3 = 1\)
This is because the direction cosines form a unit vector in three-dimensional space, and the sum of the squares of the direction cosines is always equal to 1.
Therefore, the value of \(cos^{2}θ_1 +cos^{2}θ_2+cos^{2}θ_3\) is also \(1\). (_Ans)
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Concepts Used:
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.
