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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 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:// | + | [[https:// |
- | Scientific notation also ties into [[https:// | + | Scientific notation also ties into [[https:// |
^ Symbol | ^ Symbol | ||
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*If you dot two vectors that are perfectly perpendicular, | *If you dot two vectors that are perfectly perpendicular, | ||
*If you dot two vectors that point directly opposite each other, you will get the // | *If you dot two vectors that point directly opposite each other, you will get the // | ||
- | *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 |
- | $$\vec{a}\cdot\vec{b} = \vec{b}\cdot\vec{a}$$ | + | $$\vec{a}\bullet\vec{b} = \vec{b}\bullet\vec{a}$$ |
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 $\vec{b}=\langle b_x, b_y, b_z\rangle$, | - **Using vector components** - If you have two vectors given by $\vec{a}=\langle a_x, a_y, a_z \rangle$ and $\vec{b}=\langle b_x, b_y, b_z\rangle$, | ||
- | $$\vec{a} \cdot \vec{b} = a_x b_x+a_y b_y+a_z b_z$$ | + | $$\vec{a} \bullet |
- **Using whole vectors and angles** - You can also calculate the dot product by using the magnitudes of the two vectors and the angle between them. Using this method, it is much easier to visualize taking the parallel part of one vector and multiplying by the other vector. It doesn' | - **Using whole vectors and angles** - You can also calculate the dot product by using the magnitudes of the two vectors and the angle between them. Using this method, it is much easier to visualize taking the parallel part of one vector and multiplying by the other vector. It doesn' | ||
- | $$\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| cos(\theta) = |\vec{a}|cos(\theta)|\vec{b}|$$ | + | $$\vec{a} \bullet |
=== Cross Product === | === Cross Product === | ||
+ | {{ 184_notes: | ||
+ | {{ 184_notes: | ||
+ | |||
The cross product is another way to " | The cross product is another way to " | ||
*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 " | + | *It measures " |
*If you cross two vectors that are perfectly parallel or anti-parallel, | *If you cross two vectors that are perfectly parallel or anti-parallel, | ||
*If you cross two vectors that are perfectly perpendicular, | *If you cross two vectors that are perfectly perpendicular, | ||
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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: | + | This formula comes from [[183_notes: |
- **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 Right Hand Rule, which was needed to understand [[183_notes: | + | 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: |
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- | FIXME ADD FIGURES | + |