Question bank/JEE Main/Mathematics

A line makes equal angles with the diagonals of a cube. What is
the sine of that angle?,d js[kk ?ku ds fod.kZ ds lkFk cjkcj dks.k cukrh

A$\sqrt{2/3}$B$\sqrt{1/3}$C$\sqrt{1/2}$DNone of these buesa

Show answer and solution▾

Let the side of the cube be a units: The position vectors of
points 0, A, B,C, D, E, F, G are
ekuk ?ku dh Hkqtk bdkbZ gS: fod.kZ0, A, B,C, D, E, F, G dh fLFkfr lfn'k
O: (0,0,0), A: (a, 0, 0), B: (0, a, 0), c: (0, 0, a), D: (a, a, 0),
E: (0, a, a); F: (a, 0, a), G: (a, a, a).
The Diagonals of the cube are ?ku ds fod.
and [IMAGE] where tgka
$\overrightarrow{BF} = \ a$î$- aĵ + aƙ$
Let Diagonals of the cube are ekuk ?ku dk fod.kZ [IMAGE] and vkSj[IMAGE] where tgk¡[IMAGE] $\overrightarrow{DC} = \ - a$î$- aĵ + aƙ$;
$\overrightarrow{BF} = \ a$î$- aĵ + aƙ$
[IMAGE] Let [IMAGE] be the direction unit vector of the line
which makes equal angle with all four diagonals
ekuk bdkbZ tks lHkh pkj
fod.kksZ ls cjkcj dks.k cukrk gSA
Then rc
$\frac{\overset{\rightarrow}{b} \cdot \overset{\rightarrow}{OG}}{|\overset{\rightarrow}{b} \parallel \overset{\rightarrow}{OG}|} = \frac{\overset{\rightarrow}{b} \cdot \overset{\rightarrow}{AE}}{|\overset{\rightarrow}{b} \parallel \overset{\rightarrow}{AE}|} = \frac{\overset{\rightarrow}{b} \cdot \overset{\rightarrow}{DC}}{|\overset{\rightarrow}{b} \parallel \overset{\rightarrow}{DC}|} = \frac{\overset{\rightarrow}{b} \cdot \overset{\rightarrow}{BF}}{|\overset{\rightarrow}{b} \parallel \overset{\rightarrow}{BF}|}$
$$\Rightarrow = = =
Solving these y = 0; z = $\lambda$
So the vector blfy, lfn'k xî$+ yĵ + zƙ$ A

Source: Grass-G / M-21 Mathematics Paper 19 Dec Hindi Eng.docx