184_notes:math_review

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184_notes:math_review [2017/06/05 14:23] – [Vector Multiplication] caballero184_notes:math_review [2017/08/24 16:06] – [Vector Multiplication] tallpaul
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 ===== Math Review ===== ===== Math Review =====
-The following mathematical ideas are ones that you have probably seen before - either in a math class, your mechanics physics class, or some other course. We will be relying on these ideas pretty heavily in this course, so take a few minutes to review these ideas. +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. 
  
 ==== 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, $x=5.43*10^{6} m$. +[[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, $x=5.43 \cdot 10^{6} m$. 
  
-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 $y=4.58*10^{-6}m$ 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.
  
 ^    Symbol    ^    Prefix    ^    Factor    ^    Scientific Notation    ^ ^    Symbol    ^    Prefix    ^    Factor    ^    Scientific Notation    ^
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 Two types of quantities that are particularly important for describing physical systems are scalars and vectors. Two types of quantities that are particularly important for describing physical systems are scalars and vectors.
-  * **Scalars** are quantities that can be represented by a single number (or magnitude). Typical examples of scalars include mass, time, speed, energy and volume. These quantities inherently have no direction. For example, you would never say "I have a volume of 2 m<sup>3</sup> to the left". In E&M, you will encounter new scalar quantities like electric potential. +  * **Scalars** are quantities that can be represented by a single number (or magnitude). Typical examples of scalars include mass, time, [[183_notes:displacement_and_velocity#Velocity_and_Speed|speed]], [[183_notes:define_energy|energy]] and volume. These quantities inherently have no direction. For example, you would never say "I have a volume of 2 m<sup>3</sup> to the left". In E&M, you will encounter new scalar quantities like [[184_notes:pc_potential|electric potential]]. 
-  * **Vectors** are quantities that //do// have a magnitude //and// a direction. Typical examples of vectors include displacement, velocity, force, and momentum. These quantities must have both the number and direction. For example, you could describe a car's velocity as 45 mph going east or 45 mph in the +x direction. In E&M, you will encounter new vectors like electric field. +  * **Vectors** are quantities that //do// have a magnitude //and// a direction. Typical examples of vectors include [[183_notes:displacement_and_velocity#Motion_(Changes_of_Position)|displacement]], [[183_notes:displacement_and_velocity#Velocity_and_Speed|velocity]], [[183_notes:momentum_principle#Net_Force|force]], and [[183_notes:momentum|momentum]]. These quantities must have both the number and direction. For example, you could describe a car's velocity as 20 m/s (45 mph) going east or 20 m/s in the +x direction. In E&M, you will encounter new vectors like [[184_notes:pc_efield|electric field]]. 
  
 Scalar quantities are easy to add, multiply, or divide as they are just numbers. Vector quantities require specific ways to add and multiply because they have a direction associated with them. The rest of these notes will provide a brief overview of vector math, but a more [[183_notes:scalars_and_vectors|thorough review can be found here]]. Scalar quantities are easy to add, multiply, or divide as they are just numbers. Vector quantities require specific ways to add and multiply because they have a direction associated with them. The rest of these notes will provide a brief overview of vector math, but a more [[183_notes:scalars_and_vectors|thorough review can be found here]].
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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,  
-$$\vec{a}\cdot\vec{b} = \vec{b}\cdot\vec{a}$$+$$\vec{a}\bullet\vec{b} = \vec{b}\bullet\vec{a}$$
  
 There are a couple of ways to calculate the dot product: There are a couple of ways to calculate the dot product:
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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.
-$$\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| cos(\theta) = |\vec{a}|cos(\theta)|\vec{b}|$$+$$\vec{a} \bullet \vec{b} = |\vec{a}| |\vec{b}| cos(\theta) = |\vec{a}|cos(\theta)|\vec{b}|$$
  
 === 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:crossproductb.png?125|A cross product multiplies the perpendicular parts of two vectors using the angle between them.}}
 +
 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:
    *It takes two vectors and creates a **__vector__** quantity (another name for the cross product is the vector product). This vector is perpendicular to the two vectors that you are crossing.    *It takes two vectors and creates a **__vector__** quantity (another name for the cross product is the vector product). This vector is perpendicular to the two vectors that you are crossing.
-   *It measures "perpendicular-ness". This lets us pick out how much of one vector points perpendicularly to another vector (i.e., how much force helps twist a wrench?).+   *It measures "perpendicular-ness". This lets us pick out how much of one vector points perpendicularly to another vector (i.e., [[183_notes:torque|how much force helps twist a wrench?]]).
       *If you cross two vectors that are perfectly parallel or anti-parallel, you will get zero.        *If you cross two vectors that are perfectly parallel or anti-parallel, you will get zero. 
       *If you cross two vectors that are perfectly perpendicular, you will get the magnitudes of the vectors multiplied together.        *If you cross two vectors that are perfectly perpendicular, you will get the magnitudes of the vectors multiplied together. 
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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 cross product by using the following formula: - **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 cross product by using the following formula:
 $$\vec{a} \times \vec{b} = \hat{x}\left(a_yb_z - b_ya_z\right)- \hat{y} \left(a_x b_z - b_xa_z\right) + \hat{z} \left(a_xb_y - b_xa_y\right)$$ $$\vec{a} \times \vec{b} = \hat{x}\left(a_yb_z - b_ya_z\right)- \hat{y} \left(a_x b_z - b_xa_z\right) + \hat{z} \left(a_xb_y - b_xa_y\right)$$
-This formula comes from [[183_notes:cross_product|taking the determinant of a special matrix]], which you may have seen in a previous math class. Note that the answer to $\vec{a} \times \vec{b}$ has components in the $\hat{x}$ direction, the $\hat{y}$ direction and the $\hat{z}$ direction; however, if any of the vector components of $\vec{a}$ or $\vec{b}$ are zero, many of these terms will drop out. An example of this from mechanics is [[183_notes:torque#torque|a torque in the xy-plane]].+This formula comes from [[183_notes:cross_product|taking the determinant of a special matrix]] (if you have taken Calculus 3 or linear algebra already, you may have seen this special matrix before). Note that the answer to $\vec{a} \times \vec{b}$ has components in the $\hat{x}$ direction, the $\hat{y}$ direction and the $\hat{z}$ direction; however, if any of the vector components of $\vec{a}$ or $\vec{b}$ are zero, many of these terms will drop out. An example of this from mechanics is [[183_notes:torque#torque|a torque in the xy-plane]].
  
 - **Using whole vectors and angles** - You can also calculate the magnitude of the cross product by using the magnitudes of the two vectors and the angle between them (but this does not tell you about the direction). Using this method, it is much easier to visualize taking the perpendicular part of one vector and multiplying by the other vector. - **Using whole vectors and angles** - You can also calculate the magnitude of the cross product by using the magnitudes of the two vectors and the angle between them (but this does not tell you about the direction). Using this method, it is much easier to visualize taking the perpendicular part of one vector and multiplying by the other vector.
 $$|\vec{a} \times \vec{b} |= |\vec{a}| |\vec{b}| sin(\theta)$$ $$|\vec{a} \times \vec{b} |= |\vec{a}| |\vec{b}| sin(\theta)$$
-To determine the direction of $\vec{a} \times \vec{b}$ using this method, you have to use the [[184_notes:rhr|Right Hand Rule]]. We will talk about the Right Hand Rule in detail in the second half of the semester.  +To determine the direction of $\vec{a} \times \vec{b}$ using this method, you have to use the Right Hand Rule, which was needed to understand [[183_notes:ang_momentum|angular momentum in mechanics]]. We will talk about the [[184_notes:rhr|Right Hand Rule]] in detail in the second half of this course. 
- +
-FIXME ADD FIGURES+
  • 184_notes/math_review.txt
  • Last modified: 2020/08/24 19:30
  • by dmcpadden