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184_notes:math_review [2017/08/24 16:06] – [Vector Multiplication] tallpaul | 184_notes:math_review [2020/08/24 19:30] (current) – dmcpadden | ||
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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. |
==== Scientific Notation ==== | ==== Scientific Notation ==== | ||
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==== Vector Notation ==== | ==== Vector Notation ==== | ||
- | {{ course_planning: | + | [{{ course_planning: |
- | 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 " | + | 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 " |
We can also define a vector in " | We can also define a vector in " | ||
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$$\vec{a} = |\vec{a}|\hat{a}$$ | $$\vec{a} = |\vec{a}|\hat{a}$$ | ||
- | We also use unit vectors to describe the x, y, and z coordinate directions. | + | We also use unit vectors to describe the x, y, and z coordinate directions. |
$$\vec{a} = a_x\hat{x}+a_y\hat{y}+a_z\hat{z}$$ | $$\vec{a} = a_x\hat{x}+a_y\hat{y}+a_z\hat{z}$$ | ||
$$\vec{a} = a_x\hat{i}+a_y\hat{j}+a_z\hat{k}$$ | $$\vec{a} = a_x\hat{i}+a_y\hat{j}+a_z\hat{k}$$ | ||
==== Vector Addition ==== | ==== Vector Addition ==== | ||
+ | [{{ course_planning: | ||
+ | [{{ course_planning: | ||
+ | |||
- | {{ course_planning: | ||
- | {{ course_planning: | ||
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.**// | + | //**Note: You CANNOT simply add or subtract the magnitudes.**// |
==== Vector Multiplication ==== | ==== Vector Multiplication ==== | ||
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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: | + | [{{ 184_notes: |
- | {{ 184_notes: | + | [{{ 184_notes: |
- **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 $\vec{b}=\langle b_x, b_y, b_z\rangle$, | ||
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=== Cross Product === | === Cross Product === | ||
- | {{ 184_notes: | + | [{{ 184_notes: |
- | {{ 184_notes: | + | [{{ 184_notes: |
The cross product is another way to " | The cross product is another way to " |