Key Concepts and Formulas

• Scalar Multiplication of Vectors: If $\vec{u}$ is a non-zero vector and $k$ is a non-zero scalar, then $\vec{v} = k\vec{u}$ means that $\vec{v}$ is parallel to $\vec{u}$.
  • If $k > 0$, $\vec{v}$ has the same direction as $\vec{u}$.
  • If $k < 0$, $\vec{v}$ has the opposite direction to $\vec{u}$.
• Angle Between Parallel Vectors:
  • If two non-zero vectors are parallel and point in the same direction, the angle between them is $0$ radians.
  • If two non-zero vectors are parallel and point in opposite directions, the angle between them is $\pi$ radians ($180^\circ$).
• Dot Product (for reference, though not strictly needed here): The angle $\theta$ between two non-zero vectors $\vec{a}$ and $\vec{c}$ can be found using $\cos \theta = \frac{\vec{a} \cdot \vec{c}}{|\vec{a}| |\vec{c}|}$.

Step-by-Step Solution

1. Understand the Given Relationships We are given two non-zero vectors, $\vec{a}$ and $\vec{b}$, and $\vec{c}$, with the following relationships: $\vec{a} = 8\vec{b} \quad \text{(Equation 1)}$ $\vec{c} = -7\vec{b} \quad \text{(Equation 2)}$ Our goal is to find the angle between vectors $\vec{a}$ and $\vec{c}$. To do this, we need to express one of these vectors as a scalar multiple of the other.

2. Establish a Direct Relationship Between $\vec{a}$ and $\vec{c}$ We can eliminate $\vec{b}$ from the given equations. From Equation 1, we can express $\vec{b}$ in terms of $\vec{a}$: $\vec{b} = \frac{1}{8}\vec{a}$ Now, substitute this expression for $\vec{b}$ into Equation 2: $\vec{c} = -7 \left( \frac{1}{8}\vec{a} \right)$ $\vec{c} = -\frac{7}{8}\vec{a}$ This equation shows that $\vec{c}$ is a scalar multiple of $\vec{a}$. Alternatively, we can express $\vec{a}$ in terms of $\vec{c}$ by rearranging the equation. Multiply both sides by $-\frac{8}{7}$: $\left(-\frac{8}{7}\right)\vec{c} = \left(-\frac{8}{7}\right)\left(-\frac{7}{8}\vec{a}\right)$ $\vec{a} = -\frac{8}{7}\vec{c}$

3. Determine the Angle Between $\vec{a}$ and $\vec{c}$ The relationship $\vec{a} = -\frac{8}{7}\vec{c}$ is in the form $\vec{u} = k\vec{v}$, where $\vec{u} = \vec{a}$, $\vec{v} = \vec{c}$, and the scalar $k = -\frac{8}{7}$. Since the scalar $k = -\frac{8}{7}$ is negative ($k < 0$), this implies that vector $\vec{a}$ is parallel to vector $\vec{c}$ and points in the opposite direction.

According to our key concepts, when two non-zero vectors are parallel and point in opposite directions, the angle between them is $\pi$ radians.

Common Mistakes & Tips

• Sign of the Scalar: Always pay close attention to the sign of the scalar multiplier. A positive scalar indicates vectors pointing in the same direction (angle $0$), while a negative scalar indicates vectors pointing in opposite directions (angle $\pi$).
• Non-Zero Vectors: The problem specifies non-zero vectors. If one of the vectors were zero, the concept of an angle between them would not be well-defined.
• Efficiency: When vectors are clearly scalar multiples of each other, directly using the property of scalar multiplication to determine the angle ($0$ or $\pi$) is more efficient than calculating the dot product.

Summary

We are given that $\vec{a} = 8\vec{b}$ and $\vec{c} = -7\vec{b}$. By eliminating $\vec{b}$, we found a direct relationship between $\vec{a}$ and $\vec{c}$: $\vec{a} = -\frac{8}{7}\vec{c}$. This shows that $\vec{a}$ is a negative scalar multiple of $\vec{c}$. Therefore, the vectors $\vec{a}$ and $\vec{c}$ are parallel and point in opposite directions. The angle between vectors pointing in opposite directions is $\pi$ radians.

The final answer is $\boxed{\pi}$, which corresponds to option (D).