Question

Show that no line in space can make angles π/6andπ/4 with X-axis and Y-axis.​

Answer 1

Answer:

Let, if possible, a line in space make angles π 6 and π 4 with X-axis and Y-axis. This is not possible, because cos γ is real. ∴ cos2γ cannot be negative. Hence, there is no line in space which makes angles π 6 and π 4 with X-axis and Y-axis.

Step-by-step explanation:

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Answer 2

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Let, if Possible a line in Space make

$angles \: \frac{\pi}{6} \: and \: \frac{\pi}{4} \: with \: X-axis \: and \: Y-axis$

$⇒ \alpha = \frac{\pi}{6} , \beta = \frac{\pi}{4}$

Let The line make angle Y with Z-axis

$⇒ {cos}^{2} \alpha + {cos}^{2} \beta + {cos}^{2} Y = 1$

$⇒ {cos}^{2} ( \frac{\pi}{6} ) + {cos}^{2} ( \frac{\pi}{4} ) + {cos}^{2} \gamma = 1$

$⇒( \frac{ \sqrt{3} }{2} {)}^{2} + ( \frac{1}{ \sqrt{2} } {)9}^{2} + {cos}^{2} \gamma = 1$

$⇒ {cos}^{2} \gamma = 1 - \frac{3}{2} - \frac{1}{2} = - \frac{1}{4}$

This is not Possible because Cos Y is real .

$⇒ {cos}^{2} Y \: Cannot \: be \: Negative$

Hence There is no line in space which makes

$angles \: \frac{\pi}{6} \: and \: \frac{\pi}{6}$

With X-axis and Y-axis .

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