Example of vector-scalar multiplication in R^2

Example: Scalar multiplication in (\mathbb{R}^2).

Let (\mathbf{v} = (2, 1) \in \mathbb{R}^2). Consider several scalar multiples:

Positive scalars

• (2 \cdot \mathbf{v} = 2 \cdot (2, 1) = (4, 2)) — doubles the length, same direction
• (\frac{1}{2} \cdot \mathbf{v} = (1, 0.5)) — halves the length, same direction
• (3 \cdot \mathbf{v} = (6, 3))

Negative scalars

• (-1 \cdot \mathbf{v} = (-2, -1)) — same length, opposite direction
• (-2 \cdot \mathbf{v} = (-4, -2)) — doubles the length, opposite direction

Zero scalar

• (0 \cdot \mathbf{v} = (0, 0) = \mathbf{0}) — the zero vector

Geometric interpretation

All scalar multiples of (\mathbf{v} = (2, 1)) lie on the line through the origin with slope (1/2). The scalar (a) determines:

• Magnitude: (|a\mathbf{v}| = |a| \cdot |\mathbf{v}|)
• Direction: same as (\mathbf{v}) if (a > 0), opposite if (a < 0)

The set ({a \cdot \mathbf{v} \mid a \in \mathbb{R}}) is the span of (\mathbf{v}), a 1-dimensional subspace of (\mathbb{R}^2).