To solve this problem, we want to find two vectors parallel to $\overrightarrow{QP}$ with a length of 4.

Step 1: Compute $\overrightarrow{QP}$

The vector $\overrightarrow{QP}$ is calculated as: $\overrightarrow{QP} = P - Q = (-4 - 3, 1 - (-4)) = (-7, 5).$

Step 2: Find the magnitude of $\overrightarrow{QP}$

The magnitude $\|\overrightarrow{QP}\|$ is given by: $\|\overrightarrow{QP}\| = \sqrt{(-7)^2 + 5^2} = \sqrt{49 + 25} = \sqrt{74}.$

Step 3: Normalize $\overrightarrow{QP}$

A unit vector in the direction of $\overrightarrow{QP}$ is: $\hat{u} = \frac{\overrightarrow{QP}}{\|\overrightarrow{QP}\|} = \left( \frac{-7}{\sqrt{74}}, \frac{5}{\sqrt{74}} \right).$

Step 4: Scale to length 4

To find a vector parallel to $\overrightarrow{QP}$ with length 4, scale the unit vector by 4: $\overrightarrow{v_1} = 4 \cdot \hat{u} = 4 \cdot \left( \frac{-7}{\sqrt{74}}, \frac{5}{\sqrt{74}} \right) = \left( \frac{-28}{\sqrt{74}}, \frac{20}{\sqrt{74}} \right).$

Step 5: Find the opposite direction vector

The vector in the opposite direction (also with length 4) is: $\overrightarrow{v_2} = -\overrightarrow{v_1} = \left( \frac{28}{\sqrt{74}}, \frac{-20}{\sqrt{74}} \right).$

Final Answer:

• Parallel vector with the same direction: $\left( \frac{-28}{\sqrt{74}}, \frac{20}{\sqrt{74}} \right)$.
• Parallel vector with the opposite direction: $\left( \frac{28}{\sqrt{74}}, \frac{-20}{\sqrt{74}} \right)$.

Would you like a step-by-step diagram or any clarifications?

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5 Related Questions:

1. How do you compute the magnitude of a vector in 2D or 3D space?
2. What are the properties of unit vectors, and how are they used?
3. How can you determine if two vectors are parallel?
4. What is the geometric interpretation of scaling vectors?
5. How does vector addition/subtraction affect the direction and magnitude?

Tip:

When working with vectors, always simplify radicals only when necessary for the context to avoid unnecessary complexity.