Vectors and magnitude
Introduced| English | Meaning |
|---|---|
| vectors/ˈvektəz/ | A quantity represented by both magnitude and direction, or by an ordered set of components. |
| scalar/ˈskeɪlə/ | A quantity described by magnitude alone, with no direction. |
| magnitude/ˈmæɡnɪtjuːd/ | The size or numerical value of a quantity. |
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
Vectors
resultant = a + b
Vectors add tip-to-tail; the resultant is the single arrow that replaces them.
Adding and subtracting vectors
- Add/subtract component by component: top with top, bottom with bottom.
- $\begin{pmatrix} 3 \\ 1 \end{pmatrix} + \begin{pmatrix} 2 \\ -4 \end{pmatrix} = \begin{pmatrix} 5 \\ -3 \end{pmatrix}$.
Scalar multiply: multiply both components by the number. $3\begin{pmatrix} 3 \\ 1 \end{pmatrix} = \begin{pmatrix} 9 \\ 3 \end{pmatrix}$.
a = (3, 1) and b = (2, −4). The top number of a + b is?
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3 + 2 = 5.
a = (3, 1) and b = (2, −4). The bottom number of a + b is?
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1 + (−4) = −3.
If a = (3, 1), the top number of 4a is?
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4 × 3 = 12.
To subtract vectors, you subtract each ______ separately.
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Subtract top from top and bottom from bottom, just like addition.
Magnitude 大小 (length)
- The magnitude of a vector uses Pythagoras:
- $\left|\begin{pmatrix} 3 \\ 4 \end{pmatrix}\right| = \sqrt{9 + 16} = \sqrt{25} = 5$.
Magnitude, not sum. The length of $\begin{pmatrix} 3 \\ 4 \end{pmatrix}$ is $\sqrt{3^2 + 4^2} = 5$, not $3 + 4 = 7$. You must use Pythagoras.
Adding by the triangle law: draw b from the tip of a, and a+b runs from start to finish
Find the magnitude (length) of the vector (3, 4).
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√(3² + 4²) = √25 = 5.
The magnitude of the vector (3, 4) is 3 + 4 = 7.
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Magnitude uses Pythagoras: √(3² + 4²) = 5, not the sum 3 + 4 = 7.
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.
Vector a = (2, −3). Find the vertical component of −2a.
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Multiply the vertical component: −2 × (−3) = 6.
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
- add/subtract vectors component by component; scalar multiply scales both
- $\begin{pmatrix} 3 \\ 1 \end{pmatrix} + \begin{pmatrix} 2 \\ -4 \end{pmatrix} = \begin{pmatrix} 5 \\ -3 \end{pmatrix}$
- magnitude $= \sqrt{x^2 + y^2}$, so $\begin{pmatrix} 3 \\ 4 \end{pmatrix}$ has length $5$