Given u=-4i+7j and v=28i-49j, are u and v parallel or orthogonal?

To determine whether the vectors **u** and **v** are parallel or orthogonal, we need to check their **dot product** and their **proportionality**.

### Step 1: Calculate the Dot Product
The dot product of two vectors \( \mathbf{u} = a_1 \mathbf{i} + b_1 \mathbf{j} \) and \( \mathbf{v} = a_2 \mathbf{i} + b_2 \mathbf{j} \) is given by:

\[
\mathbf{u} \cdot \mathbf{v} = a_1 \cdot a_2 + b_1 \cdot b_2
\]

For the vectors:
- \( \mathbf{u} = -4\mathbf{i} + 7\mathbf{j} \)
- \( \mathbf{v} = 28\mathbf{i} - 49\mathbf{j} \)

The dot product is:

\[
\mathbf{u} \cdot \mathbf{v} = (-4)(28) + (7)(-49)
\]

\[
\mathbf{u} \cdot \mathbf{v} = -112 - 343 = -455
\]

Since the dot product is **not zero** (\(\mathbf{u} \cdot \mathbf{v} \neq 0\)), the vectors are **not orthogonal**.

### Step 2: Check Proportionality
Two vectors are parallel if one is a scalar multiple of the other. We check if:

\[
\frac{a_2}{a_1} = \frac{b_2}{b_1}
\]

For the vectors \( \mathbf{u} = -4\mathbf{i} + 7\mathbf{j} \) and \( \mathbf{v} = 28\mathbf{i} - 49\mathbf{j} \):

\[
\frac{28}{-4} = \frac{-49}{7}
\]

\[
-7 = -7
\]

Since both ratios are equal, the vectors **u** and **v** are **parallel**.

### Conclusion
The vectors \( \mathbf{u} \) and \( \mathbf{v} \) are parallel.

Would you like any more details or have any questions?

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Here are 5 related questions for further exploration:
1. What is the significance of the dot product in vector analysis?
2. How do you find the angle between two vectors using the dot product?
3. What does it mean for vectors to be orthogonal in 3D space?
4. How can you determine if vectors are collinear using their cross product?
5. What are the geometric interpretations of parallel vectors?

**Tip:** Always check both the dot product and proportionality when determining if vectors are parallel or orthogonal.

Learn how to determine if two vectors are parallel or orthogonal using their dot product and proportionality. This problem involves vectors \( \mathbf{u} = -4\mathbf{i} + 7\mathbf{j} \) and \( \mathbf{v} = 28\mathbf{i} - 49\mathbf{j} \), exploring key concepts in vector analysis. Suitable for high school students.

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