The angle between the two planes x + y - z = 3 and 3x + 2y + z = 5 is:

Question

The angle between the two planes $x + y - z = 3$ and $3 x + 2 y + z = 5$ is:

Smart Achievers · Accepted Answer

(2)

Ans. $\cos^{- 1} \frac{2 \sqrt{42}}{21}$

Explanation:

$P_{1} = x + y - z = 3$

$P_{2} = 3 x + 2 y + z = 5$

${\vec{x}}_{1} = \hat{i} + \hat{j} - \hat{k}$

${\vec{x}}_{2} = 3 \hat{i} + 2 \hat{j} + \hat{k}$

$\cos θ =$ $\frac{| {\vec{x}}_{1} \cdot {\vec{x}}_{2} |}{| {\vec{x}}_{1} | | {\vec{x}}_{2} |}$ $= \frac{| 3 + 2 - 1 |}{\sqrt{3} \sqrt{14}}$

$= \frac{4}{\sqrt{42}} = \frac{4 \sqrt{42}}{42}$

$\cos θ = \frac{2 \sqrt{42}}{21}$

$∴$ $θ = \cos^{- 1} (\frac{2 \sqrt{42}}{21})$

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