Vector Operations

Vector Operations

Vectors can be added, subtracted and multiplied by a scalar (a number). These operations have clear geometric interpretations and can be performed using column vectors by operating component-by-component.

Addition and Subtraction

\[ \mathbf{a}+\mathbf{b}=\begin{pmatrix}a_1\a_2\end{pmatrix}+\begin{pmatrix}b_1\b_2\end{pmatrix}=\begin{pmatrix}a_1+b_1\a_2+b_2\end{pmatrix} \]

Geometrically, \( \mathbf{a}+\mathbf{b} \) is represented by placing the tail of \( \mathbf{b} \) at the head of \( \mathbf{a} \) — the resultant joins the original tail to the final head (the triangle law).

Subtraction: \( \mathbf{a}-\mathbf{b}=\mathbf{a}+(-\mathbf{b}) \). Negating a vector reverses its direction.

Scalar Multiplication

\[ k\mathbf{a}=k\begin{pmatrix}a_1\a_2\end{pmatrix}=\begin{pmatrix}ka_1\ka_2\end{pmatrix} \]

Multiplying a vector by scalar \( k \) scales its magnitude by \( |k| \) without changing direction (if \( k>0 \)) or reverses and scales (if \( k<0 \)).

Magnitude of a Vector

\[ |\mathbf{a}|=\left|\begin{pmatrix}a_1\a_2\end{pmatrix}\right|=\sqrt{a_1^2+a_2^2} \]

Parallel Vectors

Two vectors are parallel if one is a scalar multiple of the other: \( \mathbf{b}=k\mathbf{a} \) for some scalar \( k \). Parallel vectors have the same or opposite direction.

Worked Examples

\( \mathbf{p}=\begin{pmatrix}3\-1\end{pmatrix} \), \( \mathbf{q}=\begin{pmatrix}-2\4\end{pmatrix} \). Find \( \mathbf{p}+\mathbf{q} \), \( \mathbf{p}-\mathbf{q} \), \( 3\mathbf{p} \), \( 2\mathbf{p}-3\mathbf{q} \) and \( |\mathbf{p}| \).
\[ \mathbf{p}+\mathbf{q}=\begin{pmatrix}1\3\end{pmatrix} \qquad \mathbf{p}-\mathbf{q}=\begin{pmatrix}5\-5\end{pmatrix} \qquad 3\mathbf{p}=\begin{pmatrix}9\-3\end{pmatrix} \] \[ 2\mathbf{p}-3\mathbf{q}=\begin{pmatrix}6\-2\end{pmatrix}-\begin{pmatrix}-6\12\end{pmatrix}=\begin{pmatrix}12\-14\end{pmatrix} \qquad |\mathbf{p}|=\sqrt{9+1}=\sqrt{10} \]
Describe geometrically: starting at \( O \), go along \( \mathbf{a} \) then \( \mathbf{b} \). What single vector gives the same result?

The resultant is \( \mathbf{a}+\mathbf{b} \) — the vector from the start of \( \mathbf{a} \) to the end of \( \mathbf{b} \). This is the triangle law of vector addition.

Show that \( \mathbf{u}=\begin{pmatrix}4\-6\end{pmatrix} \) and \( \mathbf{v}=\begin{pmatrix}-6\9\end{pmatrix} \) are parallel. Find the scalar \( k \) such that \( \mathbf{v}=k\mathbf{u} \).
\[ \frac{-6}{4}=\frac{9}{-6}=-\frac{3}{2} \implies \mathbf{v}=-\frac{3}{2}\mathbf{u} \quad \checkmark \text{ parallel} \]

 Key Takeaways

  • Vector addition/subtraction: operate component-by-component.
  • Scalar multiplication: multiply each component by the scalar.
  • Magnitude: \( |\mathbf{a}|=\sqrt{a_1^2+a_2^2} \) (Pythagoras on the components).
  • Parallel vectors: one is a scalar multiple of the other — check the ratio of components is constant.
  • Triangle law: \( \overrightarrow{AB}+\overrightarrow{BC}=\overrightarrow{AC} \).