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 \(\overrightarrow{v} = k\overrightarrow{u}\).

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