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

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1. Identify the vectors \( \mathbf{v} = 3\mathbf{i} - 5\mathbf{j} \) and \( \mathbf{w} = 6\mathbf{i} - 10\mathbf{j} \).
2. To check if the vectors are parallel, see if one vector is a scalar multiple of the other. This means checking if there exists a scalar \( k \) such that \( \mathbf{w} = k \mathbf{v} \).
3. Compare the components: check if \( 6 = 3k \) and \( -10 = -5k \). If both equations have the same \( k \), then the vectors are parallel.
4. To check if the vectors are orthogonal (perpendicular), calculate their dot product using the formula \( \mathbf{v} \cdot \mathbf{w} = v_1 w_1 + v_2 w_2 \).
5. If the dot product equals zero, the vectors are orthogonal; if not, and they are not scalar multiples, then the vectors are neither parallel nor orthogonal.

Key Concepts

Vector Representation in Component Form

Vectors can be expressed in terms of their components along the coordinate axes, such as v = 3i - 5j, where i and j are unit vectors along the x and y axes. Understanding this form allows for straightforward calculation of vector operations like dot product and scalar multiplication.

Parallel Vectors

Two vectors are parallel if one is a scalar multiple of the other, meaning their components are proportional. For example, if w = k * v for some scalar k, then v and w point in the same or opposite directions, indicating parallelism.

Orthogonal Vectors and the Dot Product

Vectors are orthogonal if their dot product equals zero. The dot product is calculated by multiplying corresponding components and summing the results. If v · w = 0, the vectors are perpendicular, which is key to determining orthogonality.