Three Vector Spaces

This Demonstration shows addition and scalar multiplication on vectors in three vector spaces: 2D vectors, sequences, and functions. A vector space becomes an inner-product space with an inner product. In particular, the intuitive notion of orthogonality for 2D vectors, when the inner product equals 0, is shown in the possibly unintuitive settings of sequences and functions.

The fixed vector

u

is black. You can vary the red vector

v

using the slider (angle with the horizontal axis

θ

for 2D vectors and frequency

f

for sequences and functions). The gray vector is the result of the indicated vector space operation.