Parallel Vectors (DP IB Applications & Interpretation (AI): HL): Revision Note

Parallel vectors

How can I tell if two vectors are parallel?

• Two vectors are parallel if one is a scalar multiple of the other

  • This means that all components of the vector have been multiplied by a common constant (scalar)

  • For example, $\left(\begin{array}{c}1\\ 0\\ 3\end{array}\right)$ and $\left(\begin{array}{c}2\\ 0\\ 6\end{array}\right)$ are scalar multiples

• Multiplying the components of a vector by a positive scalar changes the magnitude of the vector but not the direction

  • For example, $\left(\begin{array}{c}2\\ 0\\ 6\end{array}\right)$ is double the length of $\left(\begin{array}{c}1\\ 0\\ 3\end{array}\right)$ but in the same direction

• Multiplying the components of a vector by a negative scalar reverses the direction

• You can factorise a vector to help spot if two vectors are parallel

  • $9\mathit{i}+6\mathit{j}–3\mathit{k}=3\left(3\mathit{i}+2\mathit{j}-\mathit{k}\right)$

  • $-6\mathit{i}-4\mathit{j}+2\mathit{k}=-2\left(3\mathit{i}+2\mathit{j}-\mathit{k}\right)$

    • They are scalar multiples of the same vector so they are parallel