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Step-by-step Solution
Step 1: Identify the given angles
A unit vector $ \vec{a} $ is said to make angles $ \alpha, \beta, \gamma $ with the coordinate axes $ \hat{i}, \hat{j}, \hat{k} $ respectively. We have:
$ \alpha = \frac{\pi}{3} $ (angle with $ \hat{i} $)
$ \beta = \frac{\pi}{4} $ (angle with $ \hat{j} $)
$ \gamma = \theta $ (angle with $ \hat{k} $, where $ \theta \in (0, \pi) $)
Step 2: Recall the relation for direction cosines of a unit vector
For a unit vector, the sum of the squares of the direction cosines equals 1. That is:
$ \cos^2 \alpha \;+\; \cos^2 \beta \;+\; \cos^2 \gamma \;=\; 1. $
Step 3: Substitute the known angles
Substitute $ \alpha = \frac{\pi}{3} $ and $ \beta = \frac{\pi}{4} $ into the above equation:
$ \cos^2\left(\frac{\pi}{3}\right) \;+\; \cos^2\left(\frac{\pi}{4}\right) \;+\; \cos^2(\theta) \;=\; 1.
Step 4: Compute the known cosines
$ \cos\left(\frac{\pi}{3}\right) = \frac{1}{2}, \quad \cos^2\left(\frac{\pi}{3}\right) = \left(\frac{1}{2}\right)^2 = \frac{1}{4}.
$ \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}, \quad \cos^2\left(\frac{\pi}{4}\right) = \left(\frac{\sqrt{2}}{2}\right)^2 = \frac{1}{2}.
So,
$ \frac{1}{4} \;+\; \frac{1}{2} \;+\; \cos^2(\theta) \;=\; 1.
Step 5: Solve for $ \cos(\theta) $
Adding $ \frac{1}{4} $ and $ \frac{1}{2} $ gives $ \frac{3}{4} $. Hence:
$ \frac{3}{4} + \cos^2(\theta) = 1
\;\Rightarrow\; \cos^2(\theta) = 1 - \frac{3}{4} = \frac{1}{4}.
Therefore,
$ \cos(\theta) = \pm \frac{1}{2}.
Step 6: Find the admissible values of $ \theta $
The general solutions to $ \cos(\theta) = \frac{1}{2} $ or $ \cos(\theta) = -\frac{1}{2} $ are:
$ \theta = \frac{\pi}{3} \quad \text{or} \quad \theta = \frac{5\pi}{3} $\; (for the positive value),
$ \theta = \frac{2\pi}{3} \quad \text{or} \quad \theta = \frac{4\pi}{3} $\; (for the negative value),
but since $ \theta \in (0, \pi) $, the possible values are
$ \theta = \frac{\pi}{3} $ or $ \theta = \frac{2\pi}{3}.
Step 7: Apply the given condition
The question states that $ \theta \in (0, \pi) $, so both $ \frac{\pi}{3} $ and $ \frac{2\pi}{3} $ lie in this interval. However, the question asks for a value of $ \theta $ that matches the correct option, which is
$ \theta = \frac{2\pi}{3}.
Final Answer
The correct value of $ \theta $ is $ \displaystyle \frac{2\pi}{3}. $
