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| 184_notes:math_review [2017/08/24 16:00] – [Scientific Notation] tallpaul | 184_notes:math_review [2020/08/24 19:30] (current) – dmcpadden |
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| | /*[[184_notes:defining_a_system|Next Page: Defining a System]]*/ |
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| ===== Math Review ===== | ===== Math Review ===== |
| The following mathematical ideas are important to understand and be able to use as we will rely on them fairly heavily in this course. These notes will provide a review of these ideas with links to more thorough resources if you feel like you need more information about a topic. | The following mathematical ideas are important to understand and to be able to use as we will rely on them fairly heavily in this course. These notes will provide a review of these ideas with links to more thorough resources if you feel like you need more information about a topic. |
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| ==== Scientific Notation ==== | ==== Scientific Notation ==== |
| [[https://en.wikipedia.org/wiki/Scientific_notation|Scientific notation]] is particularly useful to represent very large and very small numbers, which will show up in E&M frequently (e.g., charges are very small objects, but the forces they experience are very strong!). The basic form of a number in scientific notation is: $number*10^{exponent}$. For example, if you have a length of $x=5,430,000 m$ then in scientific notation, $x=5.43 \cdot 10^{6} m$. | [[https://en.wikipedia.org/wiki/Scientific_notation|Scientific notation]] is particularly useful to represent very large and very small numbers, which will show up in E&M frequently (e.g., charges are very small objects, but the forces they experience are very strong!). The basic form of a number in scientific notation is: $number \cdot 10^{exponent}$. For example, if you have a length of $x=5,430,000 m$ then in scientific notation, $x=5.43 \cdot 10^{6} m$. |
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| Scientific notation also ties into [[https://en.wikipedia.org/wiki/Unit_prefix|unit prefixes]], which are commonly used in physics. For example, we could write the length of $y=.00000458m$ as $y=4.58 \cdot 10^{-6}m$ or as $y=4.58 \mu m$ (micro-meters). The common prefix names, symbols and scientific notation are shown below. | Scientific notation also ties into [[https://en.wikipedia.org/wiki/Unit_prefix|unit prefixes]], which are commonly used in physics. For example, we could write the length of $y=.00000458m$ as $y=4.58 \cdot 10^{-6}m$ or as $y=4.58 \mu m$ (micro-meters). The common prefix names, symbols and scientific notation are shown below. |
| ==== Vector Notation ==== | ==== Vector Notation ==== |
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| {{ course_planning:course_notes:3d_vector.png?300| A position vector defined in 3D space}} | [{{ course_planning:course_notes:3d_vector.png?300| A position vector defined in 3D space}}] |
| Vectors are typically drawn as an arrow. The length of the arrow represents the magnitude of the vector, and the arrow points in the same direction as the vector. The triangle end of the arrow is typically referred to as the "head" or "tip", with the other end of the arrow being the "tail". When drawn this way, a vector can easily be moved around in space as it is the difference between the tip and the tail that defines the vector itself. The tail of the arrow has no meaning besides what we assign it, for example, the location at which the vector quantities is measured. | Vectors are typically drawn as arrows. The length of the arrow represents the magnitude of the vector, and the arrow points in the same direction as the vector. The triangle end of the arrow is typically referred to as the "head" or "tip", with the other end of the arrow being the "tail". When drawn this way, a vector can easily be moved around in space as it is the difference between the tip and the tail that defines the vector itself. The tail of the arrow has no meaning besides what we assign it, for example, the location at which the vector quantities is measured. |
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| We can also define a vector in "bracket" notation: | We can also define a vector in "bracket" notation: |
| $$\vec{a} = |\vec{a}|\hat{a}$$ | $$\vec{a} = |\vec{a}|\hat{a}$$ |
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| We also use unit vectors to describe the x, y, and z coordinate directions. This are represented by an $\hat{x}$, $\hat{y}$, and $\hat{z}$ or by an $\hat{i}$, $\hat{j}$, and $\hat{k}$. Using these coordinate unit vector, you can write any vector in terms of its components. These are common alternative ways to write vectors (as opposed to the bracket notation). | We also use unit vectors to describe the x, y, and z coordinate directions. These are represented by an $\hat{x}$, $\hat{y}$, and $\hat{z}$ or by an $\hat{i}$, $\hat{j}$, and $\hat{k}$. Using these coordinate unit vectors, you can write any vector in terms of its components. These are common alternative ways to write vectors (as opposed to the bracket notation). |
| $$\vec{a} = a_x\hat{x}+a_y\hat{y}+a_z\hat{z}$$ | $$\vec{a} = a_x\hat{x}+a_y\hat{y}+a_z\hat{z}$$ |
| $$\vec{a} = a_x\hat{i}+a_y\hat{j}+a_z\hat{k}$$ | $$\vec{a} = a_x\hat{i}+a_y\hat{j}+a_z\hat{k}$$ |
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| ==== Vector Addition ==== | ==== Vector Addition ==== |
| | [{{ course_planning:course_notes:2d_vector_addition.png?225|graphical vector addition }}] |
| | [{{ course_planning:course_notes:2d_vector_subtraction.png?225|graphical vector subtraction }}] |
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| {{ course_planning:course_notes:2d_vector_addition.png?250|graphical vector addition}} | |
| {{ course_planning:course_notes:2d_vector_subtraction.png?225|graphical vector subtraction}} | |
| Two vectors are added (or subtracted) component by component: | Two vectors are added (or subtracted) component by component: |
| $$ \vec{a} +\vec{b} = \langle a_x, a_y, a_z \rangle + \langle b_x, b_y, b_z \rangle = \langle a_x+b_x, a_y+b_y, a_z+b_z \rangle $$ | $$ \vec{a} +\vec{b} = \langle a_x, a_y, a_z \rangle + \langle b_x, b_y, b_z \rangle = \langle a_x+b_x, a_y+b_y, a_z+b_z \rangle $$ |
| $$ \vec{a} - \vec{b} = \langle a_x, a_y, a_z \rangle - \langle b_x, b_y, b_z \rangle = \langle a_x-b_x, a_y-b_y, a_z-b_z \rangle $$ | $$ \vec{a} - \vec{b} = \langle a_x, a_y, a_z \rangle - \langle b_x, b_y, b_z \rangle = \langle a_x-b_x, a_y-b_y, a_z-b_z \rangle $$ |
| //**Note: You CANNOT simply add or subtract the magnitudes.**// This disregards the direction that the vectors point in. Alternatively, you can use the [[183_notes:scalars_and_vectors#adding_&_subtracting_vectors|"tip-to-tail" method]] to add or subtract vectors if you have them drawn out graphically. | //**Note: You CANNOT simply add or subtract the magnitudes.**// This disregards the direction that the vectors point in. Alternatively, you can use the [[183_notes:scalars_and_vectors#adding_&_subtracting_vectors|"tip-to-tail" method]] to add or subtract vectors if you have them drawn out graphically. |
| ==== Vector Multiplication ==== | ==== Vector Multiplication ==== |
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| *If you dot two vectors that are perfectly perpendicular, you will get zero. | *If you dot two vectors that are perfectly perpendicular, you will get zero. |
| *If you dot two vectors that point directly opposite each other, you will get the //negative// of the magnitudes multiplied together. | *If you dot two vectors that point directly opposite each other, you will get the //negative// of the magnitudes multiplied together. |
| *The order of the dot product does not matter. This means that $\vec{a} \cdot \vec{b}$ will give you the same answer as $\vec{b} \cdot \vec{a}$, that is, | *The order of the dot product does not matter. This means that $\vec{a} \bullet \vec{b}$ will give you the same answer as $\vec{b} \bullet \vec{a}$, that is, |
| $$\vec{a}\cdot\vec{b} = \vec{b}\cdot\vec{a}$$ | $$\vec{a}\bullet\vec{b} = \vec{b}\bullet\vec{a}$$ |
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| There are a couple of ways to calculate the dot product: | There are a couple of ways to calculate the dot product: |
| {{ 184_notes:dotproducta.png?125|A dot product multiplies the parallel parts of two vectors using the angle between them.}} | [{{ 184_notes:dotproducta.png?125|A dot product multiplies the parallel parts of two vectors using the angle between them.}}] |
| {{ 184_notes:dotproductb.png?150|A dot product multiplies the parallel parts of two vectors using the angle between them.}} | [{{ 184_notes:dotproductb.png?150|A dot product multiplies the parallel parts of two vectors using the angle between them.}}] |
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| - **Using vector components** - If you have two vectors given by $\vec{a}=\langle a_x, a_y, a_z \rangle$ and $\vec{b}=\langle b_x, b_y, b_z\rangle$, then you can calculate the dot product by multiplying each component together and adding them together: | - **Using vector components** - If you have two vectors given by $\vec{a}=\langle a_x, a_y, a_z \rangle$ and $\vec{b}=\langle b_x, b_y, b_z\rangle$, then you can calculate the dot product by multiplying each component together and adding them together: |
| $$\vec{a} \cdot \vec{b} = a_x b_x+a_y b_y+a_z b_z$$ | $$\vec{a} \bullet \vec{b} = a_x b_x+a_y b_y+a_z b_z$$ |
| - **Using whole vectors and angles** - You can also calculate the dot product by using the magnitudes of the two vectors and the angle between them. Using this method, it is much easier to visualize taking the parallel part of one vector and multiplying by the other vector. It doesn't matter whether you take the part of $\vec{a}$ that is parallel to $\vec{b}$ or the part of $\vec{b}$ that is parallel to $\vec{a}$, you will get the same answer. | - **Using whole vectors and angles** - You can also calculate the dot product by using the magnitudes of the two vectors and the angle between them. Using this method, it is much easier to visualize taking the parallel part of one vector and multiplying by the other vector. It doesn't matter whether you take the part of $\vec{a}$ that is parallel to $\vec{b}$ or the part of $\vec{b}$ that is parallel to $\vec{a}$, you will get the same answer. |
| $$\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| cos(\theta) = |\vec{a}|cos(\theta)|\vec{b}|$$ | $$\vec{a} \bullet \vec{b} = |\vec{a}| |\vec{b}| cos(\theta) = |\vec{a}|cos(\theta)|\vec{b}|$$ |
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| === Cross Product === | === Cross Product === |
| {{ 184_notes:crossproducta.png?125|A cross product multiplies the perpendicular parts of two vectors using the angle between them.}} | [{{ 184_notes:crossproducta.png?125|A cross product multiplies the perpendicular parts of two vectors using the angle between them.}}] |
| {{ 184_notes:crossproductb.png?125|A cross product multiplies the perpendicular parts of two vectors using the angle between them.}} | [{{ 184_notes:crossproductb.png?125|A cross product multiplies the perpendicular parts of two vectors using the angle between them.}}] |
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| The cross product is another way to "multiply" two vectors together, which again has some important features: | The cross product is another way to "multiply" two vectors together, which again has some important features: |