1. Key Concepts and Formulas

• Dot Product of Perpendicular Vectors: Two non-zero vectors $\overrightarrow u$ and $\overrightarrow v$ are perpendicular if and only if their dot product is zero: $\overrightarrow u \cdot \overrightarrow v = 0$.
• Definition of Dot Product: The dot product of two vectors $\overrightarrow u$ and $\overrightarrow v$ is given by $\overrightarrow u \cdot \overrightarrow v = |\overrightarrow u| |\overrightarrow v| \cos \theta$, where $|\overrightarrow u|$ and $|\overrightarrow v|$ are their magnitudes, and $\theta$ is the angle between them.
• Properties of Dot Product:
  • Distributive property: $\overrightarrow u \cdot (\overrightarrow v + \overrightarrow w) = \overrightarrow u \cdot \overrightarrow v + \overrightarrow u \cdot \overrightarrow w$.
  • Scalar multiplication: $(k\overrightarrow u) \cdot \overrightarrow v = k(\overrightarrow u \cdot \overrightarrow v)$.
  • Self-dot product: $\overrightarrow u \cdot \overrightarrow u = |\overrightarrow u|^2$.
• Unit Vectors: A unit vector has a magnitude of 1. If $\widehat u$ is a unit vector, then $|\widehat u| = 1$, and $\widehat u \cdot \widehat u = |\widehat u|^2 = 1$.

2. Step-by-Step Solution

Step 1: Use the Perpendicularity Condition We are given that vectors $\overrightarrow c = \widehat a + 2\widehat b$ and $\overrightarrow d = 5\widehat a - 4\widehat b$ are perpendicular. By the definition of perpendicular vectors, their dot product must be zero. $\overrightarrow c \cdot \overrightarrow d = 0$ Substituting the given expressions for $\overrightarrow c$ and $\overrightarrow d$: $(\widehat a + 2\widehat b) \cdot (5\widehat a - 4\widehat b) = 0$

Step 2: Expand the Dot Product We expand the dot product using the distributive property, similar to multiplying binomials. $\widehat a \cdot (5\widehat a) + \widehat a \cdot (-4\widehat b) + (2\widehat b) \cdot (5\widehat a) + (2\widehat b) \cdot (-4\widehat b) = 0$ Using the property of scalar multiplication and the commutative property of the dot product ($\widehat a \cdot \widehat b = \widehat b \cdot \widehat a$): $5(\widehat a \cdot \widehat a) - 4(\widehat a \cdot \widehat b) + 10(\widehat b \cdot \widehat a) - 8(\widehat b \cdot \widehat b) = 0$ $5(\widehat a \cdot \widehat a) - 4(\widehat a \cdot \widehat b) + 10(\widehat a \cdot \widehat b) - 8(\widehat b \cdot \widehat b) = 0$

Step 3: Simplify Using Unit Vector Properties We are given that $\overrightarrow a$ and $\overrightarrow b$ are unit vectors. This means $|\overrightarrow a| = 1$ and $|\overrightarrow b| = 1$. Therefore, their self-dot products are:

• $\widehat a \cdot \widehat a = |\widehat a|^2 = 1^2 = 1$
• $\widehat b \cdot \widehat b = |\widehat b|^2 = 1^2 = 1$

Substitute these values into the expanded equation: $5(1) - 4(\widehat a \cdot \widehat b) + 10(\widehat a \cdot \widehat b) - 8(1) = 0$ Combine the constant terms and the terms involving $\widehat a \cdot \widehat b$: $(5 - 8) + (-4 + 10)(\widehat a \cdot \widehat b) = 0$ $-3 + 6(\widehat a \cdot \widehat b) = 0$

Step 4: Solve for the Dot Product $\widehat a \cdot \widehat b$ Rearrange the equation to solve for $\widehat a \cdot \widehat b$: $6(\widehat a \cdot \widehat b) = 3$ $\widehat a \cdot \widehat b = \frac{3}{6}$ $\widehat a \cdot \widehat b = \frac{1}{2}$

Step 5: Find the Angle Between $\overrightarrow a$ and $\overrightarrow b$ Now, we use the definition of the dot product: $\widehat a \cdot \widehat b = |\widehat a| |\widehat b| \cos \theta$, where $\theta$ is the angle between $\overrightarrow a$ and $\overrightarrow b$. Since $\widehat a$ and $\widehat b$ are unit vectors, $|\widehat a| = 1$ and $|\overrightarrow b| = 1$. $\frac{1}{2} = (1)(1) \cos \theta$ $\cos \theta = \frac{1}{2}$ We need to find the angle $\theta$ such that $\cos \theta = \frac{1}{2}$ and $0 \le \theta \le \pi$. This angle is $\theta = \frac{\pi}{3}$.

3. Common Mistakes & Tips

• Tip: Always remember that for unit vectors $\widehat u$ and $\widehat v$, $\widehat u \cdot \widehat u = 1$ and $\widehat v \cdot \widehat v = 1$. This simplifies calculations significantly.
• Mistake: Errors in expanding the dot product are common. Treat it like algebraic multiplication, carefully distributing each term and paying attention to signs.
• Tip: When solving $\cos \theta = \frac{1}{2}$, recall the standard trigonometric values. The principal value for $\theta$ in the range $[0, \pi]$ is $\frac{\pi}{3}$.

4. Summary

The problem required us to find the angle between two unit vectors $\overrightarrow a$ and $\overrightarrow b$, given that two linear combinations of these vectors, $\overrightarrow c$ and $\overrightarrow d$, are perpendicular. We utilized the property that the dot product of perpendicular vectors is zero. By expanding the dot product $(\widehat a + 2\widehat b) \cdot (5\widehat a - 4\widehat b) = 0$ and using the fact that $\overrightarrow a$ and $\overrightarrow b$ are unit vectors (meaning $\widehat a \cdot \widehat a = 1$ and $\widehat b \cdot \widehat b = 1$), we simplified the equation to find $\widehat a \cdot \widehat b = \frac{1}{2}$. Finally, using the definition of the dot product $\widehat a \cdot \widehat b = |\widehat a| |\widehat b| \cos \theta$, and knowing $|\widehat a|=|\widehat b|=1$, we solved for $\cos \theta = \frac{1}{2}$, which gives $\theta = \frac{\pi}{3}$.

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