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math:vectors [2021/03/08 03:43] – created epixmath:vectors [2021/03/18 02:56] (current) – added a note on graphing vectors epix
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 ====== Vectors ====== ====== Vectors ======
-Vectors are line segments that represent the magnitude and direction of a quantity, which is useful for real life quantities like force and velocity. It is usually described by two points/ordered pairs. Vectors are written with an arrow on top of the two point letters instead of a line like for line segments.+Vectors are line segments that represent the magnitude and direction of a quantity, which is useful for real life quantities like force and velocity. It is usually described by two points/ordered pairs. Vectors are written with an arrow on top of the two point letters instead of a line like for line segments. Sometimes it may be useful to graph two vectors as the sum of their parts. Doing this allows you to do trigonometry to find values of the combined vector on harder problems without the hassle of vector math. Trigonometry is also indefinitely useful for this unit. 
 ===== Component Form ===== ===== Component Form =====
 The component form of a vector is the individual changes in horizontal and vertical units that make up the combined vector's magnitude and direction. If the vector is graphed, just //count the boxes™//, otherwise do $y_2 - y_1$ and $x_2 - x_1$ with the ordered pairs they give you for each component. Component form is written with angled brackets (e.g. <2,-1>) to distinguish it from standard ordered pairs. The component form of a vector is the individual changes in horizontal and vertical units that make up the combined vector's magnitude and direction. If the vector is graphed, just //count the boxes™//, otherwise do $y_2 - y_1$ and $x_2 - x_1$ with the ordered pairs they give you for each component. Component form is written with angled brackets (e.g. <2,-1>) to distinguish it from standard ordered pairs.
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 ===== Direction ===== ===== Direction =====
-The direction of the vector is pretty self-explanatory. It can be found by doing some basic trig on the vector, with theta being the angle at the origin point and between the hypotenuse and adjacent side. Again, think of the vector as a right angle triangle.+The direction of the vector is pretty self-explanatory. It can be found by doing some basic trig on the vector, with theta being the angle at the origin point and between the hypotenuse and adjacent side. Again, think of the vector as a right angle triangle((hint: use $\tan$)).
  
 ===== Math with Vectors ===== ===== Math with Vectors =====
 Math with vectors is super simple! To add two vectors together, just add their component values together((i.e. <1,1> + <2,-3> = <3,-2>)). For multiplication with a scalar value, just multiply in the value to both horizontal and vertical components ((i.e. 2<1,1> = <2*1,2*1> = <2,2>)). To subtract, scalar multiply the second value by -1 and then add. Math with vectors is super simple! To add two vectors together, just add their component values together((i.e. <1,1> + <2,-3> = <3,-2>)). For multiplication with a scalar value, just multiply in the value to both horizontal and vertical components ((i.e. 2<1,1> = <2*1,2*1> = <2,2>)). To subtract, scalar multiply the second value by -1 and then add.
 +
 +==== Dot Product ====
 +The dot product of two vectors is just multiplying the two together. The answer is a scalar value. Take the vectors <x, y> and <j, k>. The //dot product// of the two would be $xj + yk$. Dot product is useful to find the angle between two vectors.
 +
 +===== Angle Between Two Vectors =====
 +Take the $\arccos$ of the dot product of the two vectors divided by the magnitudes of the vectors multiplied together. In other words, if vectors $v$ and $w$ exist, the angle $\theta$ between them would be $\theta = \arccos {\frac{v \cdot w}{||v||||w||}}$. The alternative equation for this would be $v \cdot w = ||v||||w||\cos \theta$.
  
 ===== Special Vectors ===== ===== Special Vectors =====
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 The zero vector has its origin at (0,0) and end at (0,0). Its variable name is a bold zero((**0**)) and its component form would also be <0,0>. The zero vector has its origin at (0,0) and end at (0,0). Its variable name is a bold zero((**0**)) and its component form would also be <0,0>.
 ==== Unit Vector ==== ==== Unit Vector ====
-A unit vector is a vector with magnitude 1. To normalize a vector to a unit vector, scalar divide((reciprocal scalar multiply)) the vector by its magnitude.+A unit vector is a vector with magnitude 1. To normalize a vector to a unit vector, scalar divide((reciprocal scalar multiply)) the vector by its magnitude. Unit vectors can also be made as <$\cos \theta$, $\sin \theta$> where $\theta$ is the angle of the vector. If you multiply unit vectors by the magnitude, you get the vector itself((wooooah! who would've known!!!! its not like that's **literally** the point of a unit vector... btw this is trig form if you are ever asked to write a vector that way, just multiply both trig functions by the magnitude.)). 
 ==== Standard Unit Vectors ==== ==== Standard Unit Vectors ====
 There are two: There are two:
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   * **j** = <0,1>   * **j** = <0,1>
 As these are the the base horizontal and vertical component forms, they can be added together to make any integer vector. For example, <3,2> can be written as 3i + 2j. Sometimes vectors are given to you in terms of **i** and **j**. To add these types of vectors together, just substitute them with **i** and **j** and simplify via algebra. As these are the the base horizontal and vertical component forms, they can be added together to make any integer vector. For example, <3,2> can be written as 3i + 2j. Sometimes vectors are given to you in terms of **i** and **j**. To add these types of vectors together, just substitute them with **i** and **j** and simplify via algebra.
 +
 +===== Parallel or Orthogonal =====
 +Parallel vectors are vector with the same direction but varying magnitude. Check if dividing the two vector's component values give the same ratio for each to see if it is parallel or not. Orthogonal vectors are vectors with a 90 degree angle between them. To check if two vectors are orthogonal, check if their **dot product** equals zero, if so it is orthogonal.

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