Vectors and magnitude · ⁨向量与模⁩

Arrows with attitude

• Wind speed, force, velocity — they all have a size and a direction. That's what makes them vectors 向量.
• A scalar 标量 (like temperature) has only size. A vector needs both.

Writing vectors

• A vector is written as a column $\begin{pmatrix} x \\ y \end{pmatrix}$, as $\overrightarrow{AB}$, or in bold $\mathbf{a}$.
• $x$ is the horizontal component (right is positive), $y$ is the vertical component (up is positive).

Forces like wind are vectors, with both size and direction

Adding and subtracting vectors

• Add/subtract component by component: top with top, bottom with bottom.
• $\begin{pmatrix} 3 \\ 1 \end{pmatrix} + \begin{pmatrix} 2 \\ -4 \end{pmatrix} = \begin{pmatrix} 5 \\ -3 \end{pmatrix}$.

Magnitude 大小 (length)

• The magnitude of a vector uses Pythagoras:

• $\left|\begin{pmatrix} 3 \\ 4 \end{pmatrix}\right| = \sqrt{9 + 16} = \sqrt{25} = 5$.

Adding by the triangle law: draw b from the tip of a, and a+b runs from start to finish

Reverse a direction

• If a vector takes you from A to B, its negative takes you from B to A.
• For $\overrightarrow{AB}=\begin{pmatrix}3\\4\end{pmatrix}$, the reverse is $\overrightarrow{BA}=\begin{pmatrix}-3\\-4\end{pmatrix}$. Both have magnitude 5.

Scalar multiplication and a zero check

• Multiply both components: if $\mathbf a=\begin{pmatrix}2\\-3\end{pmatrix}$, then $3\mathbf a=\begin{pmatrix}6\\-9\end{pmatrix}$ and $-\mathbf a=\begin{pmatrix}-2\\3\end{pmatrix}$. A negative multiplier reverses direction.
• Magnitude remains nonnegative: $|3\mathbf a|=3\sqrt{13}$; the zero vector has magnitude 0. Nonzero parallel vectors are scalar multiples.

You've got it