The angle between any pair of diagonals can be shown to be Cosθ=1/3.
The correct answer is Option C) \(\cos^{-1} \left(\frac{1}{3} \right)\)
O(0, 0, 0), A(a, 0, 0), B(0, a, 0), R(0, 0, a), D(a, a, 0), K(a, 0, a), L(0, a, a,), P(a, a, a) direction-cosines of diagonal Op is proportional to (a, a, a)
so, Op's direct cosines are:
The angle between two diagonals of a cube is cos–11/3.
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The correct answer is Option C) \(\cos^{-1} \left(\frac{1}{3} \right)\)
The correct answer is Option C) \(\cos^{-1} \left(\frac{1}{3} \right)\)
DC’s of diagonals OP and AQ are
1/√3,1/√3,1/√3and−1/√3,1/√3,1/√3
So, cos θ=(1/√3)(−1/√3)+(1/√3)(1/√3)+(1/√3)(1/√3)
=1/3
so, θ = cos–1 (1/3).
The Diagonal of a Cube Formula can be denoted by:
The diagonal of a Cube = \(\begin{array}{l}\sqrt{3}x\end{array}\)
The primary diagonal of a Cube cuts through the center of the Cube; the diagonal of the Cube's face is not the main diagonal. The main diagonal of a cube can be determined by the help of multiplying the length of one side with the square root of 3 (it is also called the body diagonal of a cube).
Also Read:
List - I | List - II | ||
(P) | γ equals | (1) | \(-\hat{i}-\hat{j}+\hat{k}\) |
(Q) | A possible choice for \(\hat{n}\) is | (2) | \(\sqrt{\frac{3}{2}}\) |
(R) | \(\overrightarrow{OR_1}\) equals | (3) | 1 |
(S) | A possible value of \(\overrightarrow{OR_1}.\hat{n}\) is | (4) | \(\frac{1}{\sqrt6}\hat{i}-\frac{2}{\sqrt6}\hat{j}+\frac{1}{\sqrt6}\hat{k}\) |
(5) | \(\sqrt{\frac{2}{3}}\) |
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.
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.