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183_notes:scalars_and_vectors [2018/05/29 19:36] – hallstein | 183_notes:scalars_and_vectors [2021/01/28 21:17] – [Adding & Subtracting Vectors] 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: | + | {{ basic_vector_new.png?300| }} |
- | //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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==== Determining Vector Components in Two Dimensions ==== | ==== Determining Vector Components in Two Dimensions ==== | ||
- | [{{ course_planning:course_notes: | + | [{{ 183_notes:2d_vector_new.png?250|2D vector decomposition into components}}] |
Two dimensional vectors are easy to sketch, so often we will use them when describing different physical systems and problems. For these vectors, it is often useful to define an angle ($\theta$) between the vector and one of the coordinate directions (see the figure to the right). The typical relationship between the x and y components of a 2D vector and its magnitude and this angle (when defined from the positive x-axis) is: | Two dimensional vectors are easy to sketch, so often we will use them when describing different physical systems and problems. For these vectors, it is often useful to define an angle ($\theta$) between the vector and one of the coordinate directions (see the figure to the right). The typical relationship between the x and y components of a 2D vector and its magnitude and this angle (when defined from the positive x-axis) is: | ||
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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 ===== | ||
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$$ \vec{a} - \vec{b} = \langle a_x, a_y, a_z \rangle - \langle b_x, b_y, b_z \rangle = \langle a_x-b_x, a_y-b_y, a_z-b_z \rangle $$ | $$ \vec{a} - \vec{b} = \langle a_x, a_y, a_z \rangle - \langle b_x, b_y, b_z \rangle = \langle a_x-b_x, a_y-b_y, a_z-b_z \rangle $$ | ||
- | {{ course_planning:course_notes: | + | {{ 183_notes:vector_addition_new.png? |
Graphically, | Graphically, | ||
**Vector addition** For addition, place the tail of the second vector at the tip of the first vector. The vector that points from the tail of the first to the tip of the second is the sum or the " | **Vector addition** For addition, place the tail of the second vector at the tip of the first vector. The vector that points from the tail of the first to the tip of the second is the sum or the " | ||
- | {{ course_planning:course_notes: | + | {{ 183_notes:vector_subtraction_new.png? |
**Vector subtraction** For subtraction, | **Vector subtraction** For subtraction, | ||