Key Concepts and Formulas

1. Vector Triple Product (VTP) Formula: For any three vectors $\overrightarrow P, \overrightarrow Q, \overrightarrow R$, the vector triple product $(\overrightarrow P \times \overrightarrow Q) \times \overrightarrow R$ can be expanded as: $(\overrightarrow P \times \overrightarrow Q) \times \overrightarrow R = (\overrightarrow P \cdot \overrightarrow R)\overrightarrow Q - (\overrightarrow Q \cdot \overrightarrow R)\overrightarrow P$
2. Properties of Non-Collinear Vectors: If $\overrightarrow a$ and $\overrightarrow b$ are non-collinear vectors, and $k_1 \overrightarrow a + k_2 \overrightarrow b = \overrightarrow 0$ for scalars $k_1, k_2$, then $k_1 = 0$ and $k_2 = 0$.
3. Dot Product Definition: The dot product of two vectors $\overrightarrow u$ and $\overrightarrow v$ is given by $\overrightarrow u \cdot \overrightarrow v = |\overrightarrow u||\overrightarrow v|\cos\phi$, where $\phi$ is the angle between them.
4. Trigonometric Identity: $\sin^2\theta + \cos^2\theta = 1$.

Step-by-Step Solution

Step 1: Apply the Vector Triple Product Formula to the Left Side We are given the equation $(\overrightarrow a \times \overrightarrow b) \times \overrightarrow c = \frac{1}{3}|\overrightarrow b||\overrightarrow c|\overrightarrow a$. We will use the vector triple product formula $(\overrightarrow P \times \overrightarrow Q) \times \overrightarrow R = (\overrightarrow P \cdot \overrightarrow R)\overrightarrow Q - (\overrightarrow Q \cdot \overrightarrow R)\overrightarrow P$. In our case, $\overrightarrow P = \overrightarrow a$, $\overrightarrow Q = \overrightarrow b$, and $\overrightarrow R = \overrightarrow c$. Applying the formula, we get: $(\overrightarrow a \times \overrightarrow b) \times \overrightarrow c = (\overrightarrow a \cdot \overrightarrow c)\overrightarrow b - (\overrightarrow b \cdot \overrightarrow c)\overrightarrow a$

Step 2: Equate the Expanded VTP with the Right Side of the Given Equation Now we equate the result from Step 1 with the given right-hand side of the equation: $(\overrightarrow a \cdot \overrightarrow c)\overrightarrow b - (\overrightarrow b \cdot \overrightarrow c)\overrightarrow a = \frac{1}{3}|\overrightarrow b||\overrightarrow c|\overrightarrow a$

Step 3: Rearrange the Equation to Group Terms with $\overrightarrow a$ and $\overrightarrow b$ To utilize the property of non-collinear vectors, we rearrange the equation so that all terms are on one side, and we can identify coefficients of $\overrightarrow a$ and $\overrightarrow b$. $(\overrightarrow a \cdot \overrightarrow c)\overrightarrow b - (\overrightarrow b \cdot \overrightarrow c)\overrightarrow a - \frac{1}{3}|\overrightarrow b||\overrightarrow c|\overrightarrow a = \overrightarrow 0$ $(\overrightarrow a \cdot \overrightarrow c)\overrightarrow b - \left( (\overrightarrow b \cdot \overrightarrow c) + \frac{1}{3}|\overrightarrow b||\overrightarrow c| \right)\overrightarrow a = \overrightarrow 0$

Step 4: Use the Property of Non-Collinear Vectors We are given that $\overrightarrow a$ and $\overrightarrow b$ are non-zero and non-collinear. This means that if a linear combination of $\overrightarrow a$ and $\overrightarrow b$ equals the zero vector, then the coefficients of $\overrightarrow a$ and $\overrightarrow b$ must both be zero. From the equation in Step 3, the coefficient of $\overrightarrow b$ is $(\overrightarrow a \cdot \overrightarrow c)$, and the coefficient of $\overrightarrow a$ is $-\left( (\overrightarrow b \cdot \overrightarrow c) + \frac{1}{3}|\overrightarrow b||\overrightarrow c| \right)$. Since $\overrightarrow a$ and $\overrightarrow b$ are non-collinear, we must have: Coefficient of $\overrightarrow b = 0 \implies \overrightarrow a \cdot \overrightarrow c = 0$ Coefficient of $\overrightarrow a = 0 \implies (\overrightarrow b \cdot \overrightarrow c) + \frac{1}{3}|\overrightarrow b||\overrightarrow c| = 0$

Step 5: Analyze the Condition $\overrightarrow a \cdot \overrightarrow c = 0$ The condition $\overrightarrow a \cdot \overrightarrow c = 0$ implies that vector $\overrightarrow a$ is perpendicular to vector $\overrightarrow c$. This is because $\overrightarrow a$ and $\overrightarrow c$ are non-zero vectors.

Step 6: Analyze the Condition $(\overrightarrow b \cdot \overrightarrow c) + \frac{1}{3}|\overrightarrow b||\overrightarrow c| = 0$ We can rewrite this equation as: $\overrightarrow b \cdot \overrightarrow c = -\frac{1}{3}|\overrightarrow b||\overrightarrow c|$ Now, we use the definition of the dot product, $\overrightarrow b \cdot \overrightarrow c = |\overrightarrow b||\overrightarrow c|\cos\theta$, where $\theta$ is the angle between vectors $\overrightarrow b$ and $\overrightarrow c$. Substituting this into the equation: $|\overrightarrow b||\overrightarrow c|\cos\theta = -\frac{1}{3}|\overrightarrow b||\overrightarrow c|$ Since $\overrightarrow b$ and $\overrightarrow c$ are non-zero vectors, $|\overrightarrow b| \neq 0$ and $|\overrightarrow c| \neq 0$. We can divide both sides by $|\overrightarrow b||\overrightarrow c|$: $\cos\theta = -\frac{1}{3}$

Step 7: Calculate $\sin\theta$ using the Trigonometric Identity We need to find a value of $\sin\theta$. We know that $\sin^2\theta + \cos^2\theta = 1$. Substituting the value of $\cos\theta$: $\sin^2\theta + \left(-\frac{1}{3}\right)^2 = 1$ $\sin^2\theta + \frac{1}{9} = 1$ $\sin^2\theta = 1 - \frac{1}{9}$ $\sin^2\theta = \frac{8}{9}$ Taking the square root of both sides: $\sin\theta = \pm\sqrt{\frac{8}{9}}$ $\sin\theta = \pm\frac{\sqrt{8}}{\sqrt{9}}$ $\sin\theta = \pm\frac{2\sqrt{2}}{3}$

Step 8: Select the Correct Option The possible values for $\sin\theta$ are $\frac{2\sqrt{2}}{3}$ and $-\frac{2\sqrt{2}}{3}$. Looking at the given options: (A) $\frac{2}{3}$ (B) $-\frac{2\sqrt{3}}{3}$ (C) $\frac{2\sqrt{2}}{3}$ (D) $-\frac{\sqrt{2}}{3}$

Our calculated value $\frac{2\sqrt{2}}{3}$ matches option (C).

Common Mistakes & Tips

• Order of Vectors in VTP: Be careful with the order of vectors in the vector triple product formula. The formula $(\overrightarrow P \times \overrightarrow Q) \times \overrightarrow R$ is different from $\overrightarrow P \times (\overrightarrow Q \times \overrightarrow R)$.
• Non-Collinearity Condition: The key to solving this problem is recognizing that if $\overrightarrow a$ and $\overrightarrow b$ are non-collinear, then their coefficients in a linear combination that equals the zero vector must be zero.
• Sign of $\sin\theta$: The problem asks for "a value of $\sin\theta$". Since $\sin^2\theta = \frac{8}{9}$, there are two possible values for $\sin\theta$: $\frac{2\sqrt{2}}{3}$ and $-\frac{2\sqrt{2}}{3}$. The options provided include one of these.

Summary

The problem involves a vector triple product equation. By applying the vector triple product formula and using the property that non-collinear vectors have zero coefficients in a null linear combination, we derived two conditions. The condition $\overrightarrow a \cdot \overrightarrow c = 0$ implies $\overrightarrow a$ is perpendicular to $\overrightarrow c$. The second condition, involving the dot product of $\overrightarrow b$ and $\overrightarrow c$, allowed us to find $\cos\theta = -\frac{1}{3}$. Using the trigonometric identity $\sin^2\theta + \cos^2\theta = 1$, we found $\sin^2\theta = \frac{8}{9}$, leading to $\sin\theta = \pm\frac{2\sqrt{2}}{3}$. The option $\frac{2\sqrt{2}}{3}$ is present in the choices.

The final answer is \boxed{\frac{2\sqrt{2}}{3}} which corresponds to option (C).