Question

A line makes angles A, B,C,D with the four diagonals of a cube show that
Cos^2A+cos^2B+cos^2C+cos^2c+cos^2D=4/3

Answer 1

What is this I can't understand

Answer 2

do you have a way of doing this... and are just stuck at some point?

i think i could prove this using vectors ... but would like to know what method you might be trying?

Answer 3

let 2 of the diagonals be in the XY plane an the other two in the YZ planes. These two planes intersect at the Y-axis. and all 4 diagonals meet at the origin.

in the XY-plane the diagonals are: \[y=\frac1{\sqrt2}x\]\[y=-\frac1{\sqrt2}x\]


and similarly in the YZ plane, the diagonals are \[y=\frac1{\sqrt2}z\]\[y=-\frac1{\sqrt2}z\]

Answer 4

Since all lines pass through the origin, we can represent them using Position Vectors of points on those lines.

in the XY plane,
we can replace the line \(y={1\over \sqrt2}x\) with the vector \(\left< \sqrt2,1,0\right>\).

Let the unknown line be represented by the UNIT-vector \(\left< x,y,z\right>\) ... {it's magnitude is 1}

\(A\) is the angle between \(\left< x,y,z\right>\) and \(\left< \sqrt2,1,0\right>\).

\[\large \cos A={\sqrt2\cdot x +y \over 1\cdot \sqrt3}\]\[\large \cos^2 A={2x^2 +y^2+2\sqrt2y \over 3}\]

Answer 5

\(B\) is the angle between \(\left< x,y,z\right>\) and \(\left< \sqrt2,-1,0\right>\) .... { the diagonal \(y=-x/\sqrt2 \)}
\[\large \cos^2 B={2x^2 +y^2-2\sqrt2y \over 3}\]

\[\therefore\cos^2A+\cos^2B= {4x^2+2y^2 \over 3}\]

Answer 6

\(C\) is the angle between \(\left< x,y,z\right>\) and \(\left< 0,1,\sqrt2\right>\) .... { the diagonal \(y=z/\sqrt2\) }
\[\large \cos^2 C={2z^2 +y^2+2\sqrt2yz \over 3}\]

\(D\) is the angle between \(\left< x,y,z\right>\) and \(\left< 0,-1,\sqrt2\right>\) .... { the diagonal \(y=-z/\sqrt2\) }
\[\large \cos^2 D={2z^2 +y^2-2\sqrt2yz \over 3}\]

\[\therefore\cos^2C+\cos^2D= {4z^2+2y^2 \over 3}\]

Answer 7

\[\cos^2A+\cos^2B+\cos^2C+\cos^2D\\= {4x^2+4y^2+4z^2\over 3}\\\\= {4\left( x^2+y^2+z^2 \right) \over 3}\\=\frac43\]

because by definition of unit vector \(x^2+y^2+z^2=1\)