# Multiplication of a Vector by a Scalar

## Multiplication of a Vector by a Scalar

Given a non-zero scalar k and a vector $$\overrightarrow{v}$$, the product k$$\overrightarrow{v}$$ is the vector in which:

• The length is the product of the length of $$\overrightarrow{v}$$ by the absolute value of k;
• The direction is that of $$\overrightarrow{v}$$;
• The sense is that of $$\overrightarrow{v}$$ if k > 0 or its opposite if k < 0.
• If k = 0 or if $$\overrightarrow{v}$$ = 0, then $$k\overrightarrow{v}$$ = 0.
• When writing the multiplication, and the result, the scalar always precedes the vector: 2 × $$\overrightarrow{v}$$ = 2$$\overrightarrow{v}$$.
• If $$\overrightarrow{v} = (a, b)$$, then $$k\overrightarrow{v} = k(a, b) = (ka, kb)$$.

### Properties

• If $$\overrightarrow{v} = \overrightarrow{0}$$ with $$k = 0$$, then $$k\overrightarrow{v} = \overrightarrow{0}$$;
• If $$\overrightarrow{v} ≠ \overrightarrow{0}$$ and $$k > 0$$, then $$\overrightarrow{v}$$ and $$k\overrightarrow{v}$$ have the same direction and the same sense and $$\|k\overrightarrow{v}\|$$ = $$k · \|\overrightarrow{v}\|$$, where $$\|\overrightarrow{v}\|$$ represents the norm of the vector $$\overrightarrow{v}$$;
• If $$\overrightarrow{v} ≠ \overrightarrow{0}$$ and $$k < 0$$, then $$\overrightarrow{v}$$ and $$k\overrightarrow{v}$$ have the same direction, in opposite senses and $$\|k\overrightarrow{v}\|$$ = $$−k · \|\overrightarrow{v}\|$$;
• The vectors $$\overrightarrow{u}$$ and $$\overrightarrow{v}$$ are said to be collinear if there is a non-zero real number $$k$$ for which $$\overrightarrow{v} = k\overrightarrow{u}$$.