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- | ~~NOTOC~~ | + | [[184_notes: |
===== 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 | + | The following mathematical ideas are important to understand and to be able to use as we will rely on them fairly |
==== Scientific Notation ==== | ==== Scientific Notation ==== | ||
- | [[https:// | + | [[https:// |
- | Scientific notation also ties into [[https:// | + | Scientific notation also ties into [[https:// |
^ Symbol | ^ Symbol | ||
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==== Vectors and Scalars ==== | ==== Vectors and Scalars ==== | ||
+ | |||
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< | + | * **Scalars** are quantities that can be represented by a single number (or magnitude). Typical examples of scalars include mass, time, [[183_notes: |
- | * **Vectors** are quantities that //do// have a magnitude //and// a direction. Typical examples of vectors include displacement, | + | * **Vectors** are quantities that //do// have a magnitude //and// a direction. Typical examples of vectors include |
- | Scalar quantities are easy to add, multiply, or divide | + | Scalar quantities are easy to add, multiply, or divide |
==== Vector Notation ==== | ==== Vector Notation ==== | ||
- | {{ 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 " | + | [{{ course_planning: |
+ | 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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==== Unit Vectors ==== | ==== Unit Vectors ==== | ||
+ | |||
Any vector can be multiplied or divided by a scalar quantity. When you divide a vector by its own magnitude, the result is called the "unit vector." | Any vector can be multiplied or divided by a scalar quantity. When you divide a vector by its own magnitude, the result is called the "unit vector." | ||
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==== Vector Addition ==== | ==== Vector Addition ==== | ||
+ | |||
{{ course_planning: | {{ course_planning: | ||
{{ course_planning: | {{ course_planning: | ||
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==== Vector Multiplication ==== | ==== Vector Multiplication ==== | ||
+ | |||
=== Dot Product === | === Dot Product === | ||
The dot product is one way to " | The dot product is one way to " | ||
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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 $\vec{b} \cdot \vec{a}$. | + | *The order of the dot product does not matter. This means that $\vec{a} \bullet |
+ | $$\vec{a}\bullet\vec{b} = \vec{b}\bullet\vec{a}$$ | ||
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 [[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 |
- | + | ||
- | FIXME ADD FIGURES | + |