Scalar Multiplication of a Vector

Scalar multiplication of a vector v in Rⁿ by a real number k produces a new vector u in Rⁿ, where each component of v is multiplied by k. This can be expressed as: $$ u = v \cdot k = \begin{pmatrix} v_1 \\ v_2 \\ ... \\ v_n \end{pmatrix} \cdot k = \begin{pmatrix} v_1 \cdot k \\ v_2 \cdot k \\ ... \\ v_n \cdot k \end{pmatrix} $$

This operation is also known as the dot product, scalar multiplication, or simply the product.

If (x, y) represents the endpoint of vector v, then the product kv is a vector ending at (kx, ky).

The product kv of a real number k and vector v can also be obtained by multiplying the magnitude of v by k, keeping the same direction if k > 0 or reversing it if k < 0.

Example

Consider a vector in R³ and a scalar k = 2:

$$ u = \begin{pmatrix} 3 \\ 4 \\ 1 \end{pmatrix} $$

The dot product kv results in another vector in R³, where each component is multiplied by k:

$$ k \cdot u = 2 \cdot \begin{pmatrix} 3 \\ 4 \\ 1 \end{pmatrix} = \begin{pmatrix} 2 \cdot 3 \\ 2 \cdot 4 \\ 2 \cdot 1 \end{pmatrix} = \begin{pmatrix} 6 \\ 8 \\ 2 \end{pmatrix} $$