In Exercises 45–50, determine whether v and w are parallel, orthogonal, or neither.v = 3i - 5j, w = 6i + 10j

Verified step by step guidance

1. Calculate the dot product of vectors \( \mathbf{v} \) and \( \mathbf{w} \). The dot product is given by \( \mathbf{v} \cdot \mathbf{w} = (3)(6) + (-5)(10) \).
2. Evaluate the dot product expression to determine if the vectors are orthogonal. If the dot product is zero, the vectors are orthogonal.
3. Check if the vectors are parallel by comparing their components. Two vectors are parallel if one is a scalar multiple of the other.
4. To check for parallelism, compare the ratios of the corresponding components: \( \frac{3}{6} \) and \( \frac{-5}{10} \).
5. Conclude whether the vectors are parallel, orthogonal, or neither based on the results from the previous steps.

Key Concepts

Vector Operations

Understanding vector operations is crucial for analyzing the relationship between vectors. This includes addition, subtraction, and scalar multiplication, which help in determining how vectors interact with each other. In this context, we will specifically look at the dot product to assess orthogonality and the proportionality of components to check for parallelism.

Dot Product

The dot product of two vectors provides a measure of their directional alignment. It is calculated as the sum of the products of their corresponding components. If the dot product equals zero, the vectors are orthogonal (perpendicular). For the vectors v and w, calculating the dot product will help determine if they are orthogonal.

Parallel Vectors

Two vectors are considered parallel if they point in the same or opposite directions, which can be determined by checking if one vector is a scalar multiple of the other. This means that the ratios of their corresponding components must be equal. In this case, we will analyze the components of vectors v and w to see if they maintain this proportionality.