To determine if the vectors $\overrightarrow{AB}$ and $\overrightarrow{PQ}$ are parallel and whether they point in the same direction, we first need to calculate the vectors $\overrightarrow{AB}$ and $\overrightarrow{PQ}$ from the given points:

• $A = (-2, -1)$
• $B = (5, 0)$
• $P = (0, 4)$
• $Q = (-12, 8)$

Step 1: Calculate the vectors

1. Vector $\overrightarrow{AB}$: $\overrightarrow{AB} = B - A = (5, 0) - (-2, -1) = (5 + 2, 0 + 1) = (7, 1)$

2. Vector $\overrightarrow{PQ}$: $\overrightarrow{PQ} = Q - P = (-12, 8) - (0, 4) = (-12 - 0, 8 - 4) = (-12, 4)$

Step 2: Check if the vectors are parallel

Two vectors are parallel if one is a scalar multiple of the other. That is, $\overrightarrow{PQ} = k \cdot \overrightarrow{AB}$ for some constant $k$.

To check this, we compare the components: $\frac{-12}{7} \quad \text{and} \quad \frac{4}{1}$

Clearly, $\frac{-12}{7} \neq \frac{4}{1}$, so the two vectors are not parallel.

Since $\overrightarrow{AB}$ and $\overrightarrow{PQ}$ are not scalar multiples, they are not parallel and do not point in the same direction.

Would you like more details on vector operations or have any questions?

Here are 5 related questions:

1. How do you determine if two vectors are perpendicular?
2. What is the difference between parallel and collinear vectors?
3. How can the dot product be used to find the angle between two vectors?
4. Can vectors in 3D space also be parallel, and how is it determined?
5. What is the geometric interpretation of the cross product of two vectors?

Tip: Parallel vectors share the same direction or are exact opposites, but their magnitudes might differ.