The angle between the two lines whose direction cosines are - 3 4 - 1 4 - 3 2 and - 3 4 - 1 4 - - 3 2

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Distinguish Acute Angle Bisectors and Obtuse Angle Bisectors

The angle bet...

Question

The angle between the two lines whose direction cosines are $(\frac{\sqrt{3}}{4}, \frac{1}{4}, \frac{\sqrt{3}}{2})$ and $(\frac{\sqrt{3}}{4}, \frac{1}{4}, \frac{- \sqrt{3}}{2})$

π

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\frac{π}{2}

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\frac{2 π}{3}

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\frac{π}{4}

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Solution

The correct option is D $\frac{2 π}{3}$
Angle between two lines is
$C o s θ = l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2}$
$c o s θ = \frac{\sqrt{3}}{4} \cdot \frac{\sqrt{3}}{4} + \frac{1}{4} \cdot \frac{1}{4} + \frac{\sqrt{3}}{4} (\frac{\sqrt{3}}{4})$
$\frac{3}{16} + \frac{1}{16} - \frac{- 3}{4}$
$= \frac{1}{4} - \frac{- 3}{4}$
$\Rightarrow c o s θ = \frac{- 1}{2}$
$∴ θ = \frac{2 π}{3}$

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The angle between the two lines whose direction cosines are (√34,14,√32) and (√34,14,−√32)

The correct option is D 2π3Angle between two lines isCosθ=l1l2+m1m2+n1n2cosθ=√34⋅√34+14⋅14+√34(√34)316+116−−34=14−−34⇒cosθ=−12∴θ=2π3

The angle between the two lines whose direction cosines are $(\frac{\sqrt{3}}{4}, \frac{1}{4}, \frac{\sqrt{3}}{2})$ and $(\frac{\sqrt{3}}{4}, \frac{1}{4}, \frac{- \sqrt{3}}{2})$