Question

multiple choice question
what must be true of vectors in order for them to be two - dimensional?
the vectors must be at an angle other than 0° or 180° to each other.
the vectors must be at an angle other than 0°, 90°, or 180° to each other.
the vectors must be either at 0° or 190° to each other
the vectors must be at an angle other than 90° or 180° to each other

Explanation:

In a two - dimensional vector space, for two vectors to span the space (be a basis for the two - dimensional space), they must be linearly independent.

• If two vectors are at an angle of \(0^{\circ}\) (parallel) or \(180^{\circ}\) (anti - parallel), they are linearly dependent (\(\vec{v}_2 = k\vec{v}_1\), where \(k>0\) for \(0^{\circ}\) and \(k < 0\) for \(180^{\circ}\)).
• If two vectors are at an angle of \(90^{\circ}\), they are orthogonal, but two non - parallel (non - \(0^{\circ}\) and non - \(180^{\circ}\)) vectors can also form a two - dimensional space. The key is to avoid linear dependence. Linear dependence occurs when vectors are parallel (\(0^{\circ}\) or \(180^{\circ}\))

Answer:

The vectors must be at an angle other than \(0^{\circ}\) or \(180^{\circ}\) to each other.