Key Concepts and Formulas

• Direction Cosines: For a vector $\vec{v}$ making angles $\alpha$, $\beta$, and $\gamma$ with the positive x, y, and z axes respectively, the direction cosines are $\cos \alpha$, $\cos \beta$, and $\cos \gamma$.
• Fundamental Identity of Direction Cosines: The sum of the squares of the direction cosines of any vector is always equal to 1: $\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$.
• Unit Vector Property: A unit vector has a magnitude of 1. For a unit vector, its components along the x, y, and z axes are precisely its direction cosines.

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Step-by-Step Solution

Step 1: Identify the given information and the goal. We are given a unit vector $\overrightarrow{a}$ that makes an angle of $\frac{\pi}{3}$ with $\widehat{i}$ (the x-axis), an angle of $\frac{\pi}{4}$ with $\widehat{j}$ (the y-axis), and an angle of $\theta$ with $\widehat{k}$ (the z-axis). We are also given that $\theta \in (0, \pi)$. Our goal is to find a possible value of $\theta$ from the given options.

Step 2: Relate the angles to direction cosines. Let the angles the unit vector $\overrightarrow{a}$ makes with the positive x, y, and z axes be $\alpha$, $\beta$, and $\gamma$ respectively. From the problem statement:

• $\alpha = \frac{\pi}{3}$
• $\beta = \frac{\pi}{4}$
• $\gamma = \theta$

The direction cosines are $\cos \alpha$, $\cos \beta$, and $\cos \gamma$. Since $\overrightarrow{a}$ is a unit vector, its direction cosines are equal to its components along the respective axes.

Step 3: Apply the fundamental identity of direction cosines. The fundamental property relating the direction cosines of any vector is: $\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$ This identity is crucial because it connects the three angles a vector makes with the coordinate axes. We know two of the angles, so we can use this identity to find the third.

Step 4: Substitute the known values and solve for $\cos^2 \theta$. Substitute the given angles into the identity: $\cos^2 \left(\frac{\pi}{3}\right) + \cos^2 \left(\frac{\pi}{4}\right) + \cos^2 \theta = 1$ Evaluate the cosine values:

• $\cos \left(\frac{\pi}{3}\right) = \frac{1}{2}$
• $\cos \left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}$

Substitute these values into the equation: $\left(\frac{1}{2}\right)^2 + \left(\frac{1}{\sqrt{2}}\right)^2 + \cos^2 \theta = 1$ Calculate the squares: $\frac{1}{4} + \frac{1}{2} + \cos^2 \theta = 1$ Combine the constant terms: $\frac{1}{4} + \frac{2}{4} + \cos^2 \theta = 1$ $\frac{3}{4} + \cos^2 \theta = 1$ Now, isolate $\cos^2 \theta$: $\cos^2 \theta = 1 - \frac{3}{4}$ $\cos^2 \theta = \frac{1}{4}$ This step simplifies the equation and isolates the term involving our unknown angle.

Step 5: Solve for $\cos \theta$ and determine the possible values of $\theta$. Taking the square root of both sides, we get: $\cos \theta = \pm \sqrt{\frac{1}{4}}$ $\cos \theta = \pm \frac{1}{2}$ This means there are two possibilities for $\cos \theta$: $\frac{1}{2}$ or $-\frac{1}{2}$.

We are given that $\theta \in (0, \pi)$. This interval corresponds to angles in the first and second quadrants.

• Case 1: $\cos \theta = \frac{1}{2}$ In the interval $(0, \pi)$, the angle whose cosine is $\frac{1}{2}$ is $\theta = \frac{\pi}{3}$. This value is within the allowed domain.

• Case 2: $\cos \theta = -\frac{1}{2}$ In the interval $(0, \pi)$, the angle whose cosine is $-\frac{1}{2}$ is $\theta = \frac{2\pi}{3}$. This value is also within the allowed domain. We find this by recognizing that $\cos(\pi - x) = -\cos x$, so $\cos(\pi - \frac{\pi}{3}) = \cos(\frac{2\pi}{3}) = -\frac{1}{2}$.

The problem asks for "a value of $\theta$". We have found two possible values: $\frac{\pi}{3}$ and $\frac{2\pi}{3}$.

Step 6: Compare the derived values with the given options. The given options are: (A) ${{5\pi } \over {6}}$ (B) ${{5\pi } \over {12}}$ (C) ${{2\pi } \over {3}}$ (D) ${{\pi } \over {4}}$

Our derived value $\theta = \frac{2\pi}{3}$ matches option (C).

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Common Mistakes & Tips

• Forgetting the $\pm$ sign: When solving $\cos^2 \theta = \frac{1}{4}$, it's crucial to remember that $\cos \theta$ can be either $\frac{1}{2}$ or $-\frac{1}{2}$.
• Ignoring the Domain: The constraint $\theta \in (0, \pi)$ is essential for selecting the correct angle from the possible values of $\cos \theta$. Without it, you might consider angles outside the specified range.
• Calculation Errors: Double-check the squares of fractions and the addition of fractions to avoid simple arithmetic mistakes.

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Summary

The problem leverages the fundamental property of direction cosines, which states that the sum of their squares is 1. By using the given angles the unit vector makes with the x and y axes, we set up an equation to solve for the cosine of the angle with the z-axis. After finding the possible values for $\cos \theta$, we used the given domain for $\theta$ to identify the correct angle. The value $\theta = \frac{2\pi}{3}$ was found to be a valid solution and matched one of the provided options.

The final answer is $\boxed{\text{C}}$.