Question

A line makes angles α, β, γ, δ with the four diagonals of a cube. Then cos²α + cos²β + cos²γ + cos²δ is equal to

• A. 1
• B. 4/3
• C. ¾
• D. 4/5

Answer 1

Answer: (B)

4/3

Answer 2

The direction ratios of the diagonal $\overrightarrow { \mathrm { OR } }$ are (1, 1, 1). ⇒ direction cosine are $\left( \frac { 1 } { \sqrt { 3 } } , \frac { 1 } { \sqrt { 3 } } , \frac { 1 } { \sqrt { 3 } } \right)$. Similarly direction cosine of $\overrightarrow { \mathrm { AS } }$ are and those of $\overrightarrow { \mathrm { BP } }$ and $\overrightarrow { \mathrm { CQ } }$ are

$\left( \frac { 1 } { \sqrt { 3 } } , \frac { 1 } { \sqrt { 3 } } , - \frac { 1 } { \sqrt { 3 } } \right)$ $\left( \frac { 1 } { \sqrt { 3 } } , - \frac { 1 } { \sqrt { 3 } } , \frac { 1 } { \sqrt { 3 } } \right)$ .

Let l, m, n be direction cosines of the line so that

cosα = , cosβ = $\frac { \ell - m - n } { \sqrt { 3 } }$, cosγ = ,

cosδ = $\frac { \ell - m + n } { \sqrt { 3 } }$

⇒ cos²α + cos²β + cos²γ + cos²δ ==$\frac { 4 } { 3 }$ (since l² + m² + n² = 1)