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183_notes:scalars_and_vectors [2018/05/29 19:36] – hallstein | 183_notes:scalars_and_vectors [2021/01/28 20:07] – [Definitions & Diagrams] stumptyl | ||
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==== Definitions & Diagrams ==== | ==== Definitions & Diagrams ==== | ||
- | //Scalars are quantities that can be represented by a single number. Typical examples include mass, volume, density, and speed.// | + | //**Scalars** are quantities that can be represented by a single number. Typical examples include mass, volume, density, and speed.// |
- | {{ course_planning: | + | {{ course_planning: |
- | //Vectors are quantities that have both a magnitude and direction. Typical examples include displacement, | + | //**Vectors** are quantities that have both a magnitude and direction. Typical examples include displacement, |
Vectors are often represented with arrows. The end with the triangle is the " | Vectors are often represented with arrows. The end with the triangle is the " | ||
==== Defining Vectors Mathematically ==== | ==== Defining Vectors Mathematically ==== | ||
- | {{ course_planning: | + | {{ course_planning: |
We define vectors in three dimensional space relative to some origin (where the tail of the vector is located). For example, a position vector $\vec{r}$ might defined relative to the origin of coordinates. The measures of the vector along the coordinate axes are called the vector' | We define vectors in three dimensional space relative to some origin (where the tail of the vector is located). For example, a position vector $\vec{r}$ might defined relative to the origin of coordinates. The measures of the vector along the coordinate axes are called the vector' | ||
- | $$ \mathbf{r} = \vec{r} = \langle r_x, r_y, r_z \rangle $$ | + | $$ \mathbf{r} = \vec{r} = \langle r_x, r_y, r_z \rangle $$ |
- | where $r_x$, $r_y$, and $r_z$ are the vector components in the $x$, $y$, and $z$ direction respectively. They tell you "how much" of the vector $\vec{r}$ is aligned with each coordinate direction. The vector itself is denoted either in bold face (in texts) or with an arrow above it (both texts and handwritten). | + | //where $r_x$, $r_y$, and $r_z$ are the vector components in the $x$, $y$, and $z$ direction respectively.// They tell you "how much" of the vector $\vec{r}$ is aligned with each coordinate direction. The vector itself is denoted either in bold face (in texts) or with an arrow above it (both texts and handwritten). |
In physics, we often use the symbol $\vec{r}$ to represent the position vector, that is, the location of an object with respect to another point (e.g., the origin of coordinates). | In physics, we often use the symbol $\vec{r}$ to represent the position vector, that is, the location of an object with respect to another point (e.g., the origin of coordinates). | ||
+ | \\ | ||
=== Length of a vector === | === Length of a vector === | ||
- | The magnitude (or length) of a vector is a scalar quantity. Mathematically, | + | The **magnitude** (or length) of a vector is a scalar quantity. Mathematically, |
$$r = | \vec{r} | = \sqrt{r_x^2+r_y^2+r_z^2}$$ | $$r = | \vec{r} | = \sqrt{r_x^2+r_y^2+r_z^2}$$ | ||
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This calculation simply uses the [[https:// | This calculation simply uses the [[https:// | ||
+ | \\ | ||
=== Unit vector === | === Unit vector === | ||
- | Any vector can be multiplied or divided by a scalar quantity. Often it is useful to divide a vector by its own magnitude. The result is the "unit vector." | + | Any vector can be multiplied or divided by a scalar quantity. Often it is useful to divide a vector by its own magnitude. The result is the "unit vector." |
$$ \hat{r} = \dfrac{\vec{r}}{|\vec{r}|} = \dfrac{\langle r_x, r_y, r_z \rangle}{\sqrt{r_x^2+r_y^2+r_z^2}}$$ | $$ \hat{r} = \dfrac{\vec{r}}{|\vec{r}|} = \dfrac{\langle r_x, r_y, r_z \rangle}{\sqrt{r_x^2+r_y^2+r_z^2}}$$ | ||
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$$ r_y = |\vec{r}| \sin \theta $$ | $$ r_y = |\vec{r}| \sin \theta $$ | ||
- | **The above equations only work when the vectors are decomposed with along the x and y axis as defined in the figure to the right.** Oftentimes, an angle that is given or derived cannot make use of the simple decomposition formulae above. The geometric properties of the problem will dictate which trigonometric functions are used. | + | //The above equations only work when the vectors are decomposed with along the x and y axis as defined in the figure to the right.// Oftentimes, an angle that is given or derived cannot make use of the simple decomposition formulae above. The geometric properties of the problem will dictate which trigonometric functions are used. |
==== Adding & Subtracting Vectors ===== | ==== Adding & Subtracting Vectors ===== | ||