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| 184_notes:math_review [2018/05/17 13:31] – [Vector Notation] curdemma | 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]] | /*[[184_notes:defining_a_system|Next Page: Defining a System]]*/ |
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| ===== Math Review ===== | ===== Math Review ===== |
| $$\vec{a} = |\vec{a}|\hat{a}$$ | $$\vec{a} = |\vec{a}|\hat{a}$$ |
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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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| 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: |
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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: |