Vectors and magnitude · Vecteurs et grandeur
| English | Français |
|---|---|
| vectors/ˈvektəz/ | vecteurs |
| scalar/ˈskeɪlə/ | scalaire |
| magnitude/ˈmæɡnɪtjuːd/ | grandeur |
Arrows with attitude
- Wind speed, force, velocity — they all have a size · taille 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 · Vecteurs
resultant = a + b · résultante = a + b
Vectors add tip-to-tail; the resultant is the single arrow that replaces them. · Les vecteurs s'additionnent pointe à queue ; la résultante est la flèche unique qui les remplace.
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 · les deux 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? · a = (3, 1) et b = (2, −4). Le nombre du haut de a + b est ?
3 + 2 = 5.
a = (3, 1) and b = (2, −4). The bottom number of a + b is? · a = (3, 1) et b = (2, −4). Le nombre du bas de a + b est ?
1 + (−4) = −3.
If a = (3, 1), the top number of 4a is? · Si a = (3, 1), le nombre du haut de 4a est ?
4 × 3 = 12.
To subtract vectors, you subtract each ______ separately. · Pour soustraire des vecteurs, vous soustrayez chaque ______ séparément.
Subtract top from top and bottom from bottom, just like addition. · Soustrayez le haut du haut et le bas du bas, comme pour l'addition.
Magnitude · Grandeur 大小 (length)
- The · Le 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). · Trouvez la grandeur (longueur) du vecteur (3, 4).
√(3² + 4²) = √25 = 5.
The magnitude of the vector (3, 4) is 3 + 4 = 7. · La grandeur du vecteur (3, 4) est 3 + 4 = 7.
Magnitude uses Pythagoras: √(3² + 4²) = 5, not the sum 3 + 4 = 7. · La grandeur utilise Pythagore : √(3² + 4²) = 5, pas la somme 3 + 4 = 7.
Reverse a direction
- If a vector takes you from A to B, its negative takes you from B to A.
- For · Pour $\overrightarrow{AB}=\begin{pmatrix}3\\4\end{pmatrix}$, 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. · Vecteur a = (2, −3). Trouver la composante verticale de −2a.
Multiply the vertical component: −2 × (−3) = 6. · Multiplier la composante verticale : −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 · et $-\mathbf a=\begin{pmatrix}-2\\3\end{pmatrix}$. 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 $5$