Scalar Multiplication

Given a vector $\vec{a} = \begin{pmatrix} a_1 \\ a_2 \end{pmatrix}$ and a scalar $k$, the scalar multiple is:

Multiply each component by $k$. The magnitude of the result is:

The direction depends on the sign of $k$:

• $k > 0$: $k\vec{a}$ points in the same direction as $\vec{a}$
• $k < 0$: $k\vec{a}$ points in the opposite direction to $\vec{a}$
• $k = 0$: $k\vec{a} = \vec{0}$ (zero vector)

Given a vector $\vec{a} = \begin{pmatrix} a_1 \\ a_2 \end{pmatrix}$ and a scalar $k$, the scalar multiple is:

Pause — copy the scalar multiplication rule: $k\begin{pmatrix}a_1\\a_2\end{pmatrix}=\begin{pmatrix}ka_1\\ka_2\end{pmatrix}$, with the magnitude result $|k\vec{a}|=|k|\,|\vec{a}|$ into your book.

We just saw that $k\vec{a}=\begin{pmatrix}ka_1\\ka_2\end{pmatrix}$; positive $k$ preserves direction, negative $k$ reverses it, and $|k|$ scales the magnitude. That raises a question: two vectors are parallel if and only if one is a scalar multiple of the other — how do you test this algebraically, and what does it mean when $k$ is negative? This card answers it → $\vec{b}\parallel\vec{a}$ iff $\vec{b}=k\vec{a}$ for some $k\neq0$; $k<0$ means the vectors point in opposite directions (anti-parallel).

Parallel vectors: Two non-zero vectors $\vec{a}$ and $\vec{b}$ are parallel if and only if $\vec{b} = k\vec{a}$ for some non-zero scalar $k$.

• If $k > 0$, they point in the same direction.
• If $k < 0$, they point in opposite directions.

Key algebraic laws (hold for all vectors $\vec{a}$, $\vec{b}$ and scalars $k$, $m$):

These laws make it possible to expand and simplify vector expressions, just as in algebra.

Parallel vectors: Two non-zero vectors $\vec{a}$ and $\vec{b}$ are parallel if and only if $\vec{b} = k\vec{a}$ for some non-zero scalar $k$.

Pause — copy the parallel-vector test: $\vec{b}\parallel\vec{a}\iff\vec{b}=k\vec{a}$ for some $k\neq0$; $k>0$ same direction; $k<0$ opposite direction into your book.