Vector Scalar Multiplication

What is Scalar Multiplication?

Scalar multiplication involves multiplying a vector by a scalar (a real number). This operation changes the magnitude of the vector but not its direction, unless the scalar is negative, which reverses the direction.

Properties of Scalar Multiplication

• Magnitude Change: The magnitude of the vector is multiplied by the absolute value of the scalar.
• Direction: If the scalar is positive, the direction remains the same. If negative, the direction is reversed.
• Zero Vector: Multiplying any vector by zero results in the zero vector, denoted as $\mathbf{0}$.

Mathematical Representation

If $\mathbf{v} = (x, y)$ is a vector and $k$ is a scalar, then the scalar multiplication is given by:

$$k \mathbf{v} = (kx, ky)$$

Examples

Example 1: Multiply a Vector by a Scalar

Vector: $\mathbf{v} = (3, 4)$

Scalar: $k = 2$

Result: $2 \mathbf{v} = (2 \times 3, 2 \times 4) = (6, 8)$