To determine whether the vectors $\mathbf{a} = \langle -5, 3, 7 \rangle$ and $\mathbf{b} = \langle 6, -8, 2 \rangle$ are orthogonal, parallel, or neither, we need to perform the following steps:

1. Check for orthogonality:

Two vectors are orthogonal if their dot product is zero.

The dot product of two vectors $\mathbf{a} = \langle a_1, a_2, a_3 \rangle$ and $\mathbf{b} = \langle b_1, b_2, b_3 \rangle$ is calculated as: $\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3$

For the vectors $\mathbf{a} = \langle -5, 3, 7 \rangle$ and $\mathbf{b} = \langle 6, -8, 2 \rangle$, let's compute their dot product: $\mathbf{a} \cdot \mathbf{b} = (-5)(6) + (3)(-8) + (7)(2)$ $\mathbf{a} \cdot \mathbf{b} = -30 + (-24) + 14 = -40$

Since the dot product is $-40$, which is not zero, the vectors are not orthogonal.

2. Check for parallelism:

Two vectors are parallel if one is a scalar multiple of the other. This means that there exists a scalar $k$ such that: $\mathbf{a} = k \mathbf{b}$ or equivalently, the ratios of their corresponding components should be equal: $\frac{a_1}{b_1} = \frac{a_2}{b_2} = \frac{a_3}{b_3}$ Substitute the components of $\mathbf{a}$ and $\mathbf{b}$: $\frac{-5}{6}, \quad \frac{3}{-8}, \quad \frac{7}{2}$

The ratios $\frac{-5}{6}$, $\frac{3}{-8}$, and $\frac{7}{2}$ are not equal. Therefore, the vectors are not parallel.

Conclusion:

The vectors are neither orthogonal nor parallel.

Would you like any additional details or clarifications?

Here are 5 related questions for further practice:

1. What conditions must hold for two vectors to be parallel?
2. How would you determine if two 4-dimensional vectors are orthogonal?
3. Can two vectors be both parallel and orthogonal? Why or why not?
4. How does the cross product help determine if two 3D vectors are parallel?
5. What is the geometric interpretation of the dot product of two vectors?

Tip: The dot product helps determine angles between vectors: if it's zero, the vectors are orthogonal; if positive, they point in a similar direction; and if negative, they point in opposite directions.