To determine whether the given vectors are orthogonal, parallel, or neither, we'll use the following criteria:

1. Orthogonal Vectors: Two vectors $\mathbf{a}$ and $\mathbf{b}$ are orthogonal if their dot product is zero, i.e., $\mathbf{a} \cdot \mathbf{b} = 0$.
2. Parallel Vectors: Two vectors $\mathbf{a}$ and $\mathbf{b}$ are parallel if one is a scalar multiple of the other, i.e., $\mathbf{a} = k\mathbf{b}$ for some scalar $k$.
3. Neither: If the vectors are neither orthogonal nor parallel, they are classified as neither.

Let's analyze each pair of vectors:

i. $\mathbf{a} = -5\hat{i} + 3\hat{j} + 7\hat{k}$ and $\mathbf{b} = 6\hat{i} - 8\hat{j} + 2\hat{k}$

• Dot Product: $\mathbf{a} \cdot \mathbf{b} = (-5)(6) + (3)(-8) + (7)(2) = -30 - 24 + 14 = -40$ Since $\mathbf{a} \cdot \mathbf{b} \neq 0$, they are not orthogonal.

• Parallel Check: Compare the components: $\frac{-5}{6}, \quad \frac{3}{-8}, \quad \frac{7}{2}$ These ratios are not equal, so the vectors are not parallel.

• Conclusion: Neither

ii. $\mathbf{a} = 4\hat{i} + 6\hat{j}$ and $\mathbf{b} = -3\hat{i} + 2\hat{j}$

• Dot Product: $\mathbf{a} \cdot \mathbf{b} = (4)(-3) + (6)(2) = -12 + 12 = 0$ Since $\mathbf{a} \cdot \mathbf{b} = 0$, they are orthogonal.

• Conclusion: Orthogonal

iii. $\mathbf{a} = -\hat{i} + 2\hat{j} + 5\hat{k}$ and $\mathbf{b} = 3\hat{i} + 4\hat{j} - \hat{k}$

• Dot Product: $\mathbf{a} \cdot \mathbf{b} = (-1)(3) + (2)(4) + (5)(-1) = -3 + 8 - 5 = 0$ Since $\mathbf{a} \cdot \mathbf{b} = 0$, they are orthogonal.

• Conclusion: Orthogonal

iv. $\mathbf{a} = 2\hat{i} + 6\hat{j} - 4\hat{k}$ and $\mathbf{b} = -3\hat{i} - 9\hat{j} + 6\hat{k}$

• Dot Product: $\mathbf{a} \cdot \mathbf{b} = (2)(-3) + (6)(-9) + (-4)(6) = -6 - 54 - 24 = -84$ Since $\mathbf{a} \cdot \mathbf{b} \neq 0$, they are not orthogonal.

• Parallel Check: Compare the components: $\frac{2}{-3}, \quad \frac{6}{-9}, \quad \frac{-4}{6}$ All ratios are equal to $-\frac{2}{3}$, so the vectors are parallel.

• Conclusion: Parallel

v. $\mathbf{u} = -3\hat{i} + 9\hat{j} + 6\hat{k}$ and $\mathbf{v} = 2\hat{i} - \hat{j} + \hat{k}$

• Dot Product: $\mathbf{u} \cdot \mathbf{v} = (-3)(2) + (9)(-1) + (6)(1) = -6 - 9 + 6 = -9$ Since $\mathbf{u} \cdot \mathbf{v} \neq 0$, they are not orthogonal.

• Parallel Check: Compare the components: $\frac{-3}{2}, \quad \frac{9}{-1}, \quad \frac{6}{1}$ These ratios are not equal, so the vectors are not parallel.

• Conclusion: Neither

vi. $\mathbf{u} = a\hat{i} + b\hat{j} + c\hat{k}$ and $\mathbf{v} = -b\hat{i} + a\hat{j}$

• Dot Product: $\mathbf{u} \cdot \mathbf{v} = (a)(-b) + (b)(a) + (c)(0) = -ab + ab = 0$ Since $\mathbf{u} \cdot \mathbf{v} = 0$, they are orthogonal.

• Conclusion: Orthogonal

Summary

i. Neither
ii. Orthogonal
iii. Orthogonal
iv. Parallel
v. Neither
vi. Orthogonal

Would you like more details on any of these calculations, or do you have any other questions?

Related Questions:

1. How do you find the angle between two vectors?
2. What conditions must be met for two vectors to be parallel?
3. Can vectors in different dimensions be orthogonal?
4. How does the cross product relate to orthogonal vectors?
5. What is the geometric interpretation of the dot product?

Tip: Always double-check your vector components and calculations, especially when comparing for parallelism and orthogonality.