184_notes:math_review

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184_notes:math_review [2018/05/17 13:33] – [Vector Notation] curdemma184_notes:math_review [2018/05/17 13:38] – [Vector Addition] curdemma
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 where $a_x$, $a_y$, and $a_z$ are the vector components in the $x$, $y$, and $z$ direction respectively. They tell you "how much" of the vector $\vec{a}$ is aligned with each coordinate direction. The vector itself is denoted either in bold face (typical in textbooks) or with an arrow above it.  where $a_x$, $a_y$, and $a_z$ are the vector components in the $x$, $y$, and $z$ direction respectively. They tell you "how much" of the vector $\vec{a}$ is aligned with each coordinate direction. The vector itself is denoted either in bold face (typical in textbooks) or with an arrow above it. 
  
-The magnitude (or length of a vector) is a scalar quantity and is denoted by vertical lines on either side of the vector (e.g. \vec{a}). It can be found by using the [[https://en.wikipedia.org/wiki/Pythagorean_theorem|Pythagorean theorem]] in three dimensions:+The magnitude (or length of a vector) is a scalar quantity and is denoted by vertical lines on either side of the vector. It can be found by using the [[https://en.wikipedia.org/wiki/Pythagorean_theorem|Pythagorean theorem]] in three dimensions:
 $$a = | \vec{a} | = \sqrt{a_x^2+a_y^2+a_z^2}$$ $$a = | \vec{a} | = \sqrt{a_x^2+a_y^2+a_z^2}$$
  
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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. 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?250|graphical vector addition}} +[{{ course_planning:course_notes:2d_vector_addition.png?250|graphical vector addition}}] 
-{{ course_planning:course_notes:2d_vector_subtraction.png?225|graphical vector subtraction}}+[{{ 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:
 $$ \vec{a} +\vec{b} = \langle a_x, a_y, a_z \rangle + \langle b_x, b_y, b_z \rangle = \langle a_x+b_x, a_y+b_y, a_z+b_z \rangle  $$ $$ \vec{a} +\vec{b} = \langle a_x, a_y, a_z \rangle + \langle b_x, b_y, b_z \rangle = \langle a_x+b_x, a_y+b_y, a_z+b_z \rangle  $$
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  • Last modified: 2020/08/24 19:30
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