Questions: Determine if the pairs of vectors below are "parallel", "orthogonal", or "neither". a=⟨2,-1,-5⟩ and b=⟨-4,2,10⟩ are a=⟨2,-1,-5⟩ and b=⟨4,-2,-11⟩ are

Solution Steps

To determine the relationship between pairs of vectors, we can use the following criteria:

1. Parallel Vectors: Two vectors are parallel if one is a scalar multiple of the other.
2. Orthogonal Vectors: Two vectors are orthogonal if their dot product is zero.
3. Neither: If neither of the above conditions is met, the vectors are neither parallel nor orthogonal.

For each pair of vectors, we will:

• Check if the vectors are parallel by comparing the ratios of their corresponding components.
• Calculate the dot product to check if they are orthogonal.
• Determine if they are neither based on the above checks.

To check if two vectors are orthogonal, we calculate their dot product. The dot product of vectors \(\mathbf{a}\) and \(\mathbf{b}\) is given by:

\[ \mathbf{a} \cdot \mathbf{b} = 2 \times (-4) + (-1) \times 2 + (-5) \times (-2) = -8 - 2 + 10 = 0 \]

Since the dot product is zero, the vectors are orthogonal.

Two vectors are parallel if one is a scalar multiple of the other. We check if the ratios of the corresponding components are equal:

\[ \frac{-4}{2} = -2, \quad \frac{2}{-1} = -2, \quad \frac{10}{-5} = -2 \]

Since all ratios are equal, the vectors are parallel.

First, check for parallelism by comparing the ratios of the corresponding components:

\[ \frac{4}{2} = 2, \quad \frac{-2}{-1} = 2, \quad \frac{-11}{-5} \approx 2.2 \]

The ratios are not equal, so the vectors are not parallel.

Next, check for orthogonality by calculating the dot product:

\[ \mathbf{a} \cdot \mathbf{b} = 2 \times 4 + (-1) \times (-2) + (-5) \times (-11) = 8 + 2 + 55 = 65 \]

Since the dot product is not zero, the vectors are not orthogonal.

Final Answer