Multiplication of a Vector by a Scalar
Given a non-zero scalar k and a vector [latex]\overrightarrow{v}[/latex], the product k[latex]\overrightarrow{v}[/latex] is the vector in which:
- The length is the product of the length of [latex]\overrightarrow{v}[/latex] by the absolute value of k;
- The direction is that of [latex]\overrightarrow{v}[/latex];
- The sense is that of [latex]\overrightarrow{v}[/latex] if k > 0 or its opposite if k < 0.
- If k = 0 or if [latex]\overrightarrow{v}[/latex] = 0, then [latex]k\overrightarrow{v}[/latex] = 0.
- When writing the multiplication, and the result, the scalar always precedes the vector: 2 × [latex]\overrightarrow{v}[/latex] = 2[latex]\overrightarrow{v}[/latex].
- If [latex]\overrightarrow{v} = (a, b)[/latex], then [latex]k\overrightarrow{v} = k(a, b) = (ka, kb)[/latex].
Properties
- If [latex]\overrightarrow{v} = \overrightarrow{0}[/latex] with [latex]k = 0[/latex], then [latex]k\overrightarrow{v} = \overrightarrow{0}[/latex];
- If [latex]\overrightarrow{v} ≠ \overrightarrow{0}[/latex] and [latex]k > 0[/latex], then [latex]\overrightarrow{v}[/latex] and [latex]k\overrightarrow{v}[/latex] have the same direction and the same sense and [latex]\|k\overrightarrow{v}\|[/latex] = [latex]k · \|\overrightarrow{v}\|[/latex], where [latex]\|\overrightarrow{v}\|[/latex] represents the norm of the vector [latex]\overrightarrow{v}[/latex];
- If [latex]\overrightarrow{v} ≠ \overrightarrow{0}[/latex] and [latex]k < 0[/latex], then [latex]\overrightarrow{v}[/latex] and [latex]k\overrightarrow{v}[/latex] have the same direction, in opposite senses and [latex]\|k\overrightarrow{v}\|[/latex] = [latex]−k · \|\overrightarrow{v}\|[/latex];
- The vectors [latex]\overrightarrow{u}[/latex] and [latex]\overrightarrow{v}[/latex] are said to be collinear if there is a non-zero real number [latex]k[/latex] for which [latex]\overrightarrow{v} = k\overrightarrow{u}[/latex].