#mainpage --- ## Vectors ##### *The basics* - Physicists use vectors to represent quantities that **have both a magnitude and a direction**. - Vectors are denoted using a **letter with an arrow on top**: $\small\vec{A}\space\space\vec{B}\space\space\vec{C}$ - If asked to write out a vector in ***ordered pair notation* list the x component and y component of that vector separating them with a comma.** $\small\vec{A}=\vec{A_x},\vec{A_y}$ - Visually vectors are represented as **arrows**. The length of these arrows represents the magnitude of their corresponding quantity and their direction represents the direction in which that quantity is acting. - Displacement, acceleration, velocity and all other things that specify a **magnitude in a direction are considered to be vector "quantities"**. - The position of a vector is **irrelevant**. If two vectors have the same magnitude and direction **they are considered to be equal**. - In other words, if two vectors have the same length and are parallel, they are considered to be *mathematically* equal. >[!quote] >##### *Example of acceleration vector* >![[physicsvectorexample.png|850]] ##### *Coordinate vs angle vectors* - In physics **most vector quantities are represented using a scalar quantity and an angle**. These vector's are known as "angle vectors" for obvious reasons. - The scalar quantity of an angle vector represents its magnitude and the angle denotes its direction. - ==**To convert a angle vector into a coordinate vector we must use a trigonometric function:**== $\newline$ $\text{Converting an angle vector into a coordinate vector:}$ $\huge\boxed{\;\large x=\vec{a}\cos\theta\;\;\;\;\;\;y=\vec{a}\sin\theta\;}$ $\small\color{grey}\vec{a}\text{ is the magnitude of the angle vector}$ $\newline$ - Vectors can also be represented using a set of coordinates *(x, y and z)*. Vectors that are represented using these values are known as *"coordinate vectors"*. - **==To convert a coordinate vector to an angle vector we can use the inverse tangent function to find the vector's angle and the Pythagorean theorem to find its magnitude:==** $\newline$ $\text{Converting a coordinate vector into an angle vector:}$ $\huge\boxed{\;\normalsize\text{Magnitude }=\sqrt{x^2+y^2}\;\;\;\;\;\;\text{Angle }=\,\tan^{-1}\biggl(\frac{y}{x}\biggl)\;}$ $\newline$ - Both coordinate and angle vectors are visually **represented in the same way,** using an arrow. In addition, both types of vectors are mathematically denoted in the same way, using a letter with an arrow on top. ##### *Coordinate vector math* - When adding or subtracting vectors from one another **you must split those vectors into their constituent components *(x, y and sometimes z)* and perform the operation on each component individually**. The "combined" values you get from doing this will be equal to the resulting vector's corresponding components *(in other words, the number you get from adding together 2 vector's x components will be equal to the x component of their "sum" vector)*. - In order to multiply a vector by some **single scalar value** you just multiply each of its components by that value individually. - ==Multiplying a vector by -1 allows you to reverse the direction of a vector without effecting its magnitude.== ##### *Visualizing coordinate vector math* - When adding vectors together you can visually **visually calculate the resulting vector by lining up the "head" of every constituent vector to the tail of another**. ==The "sum" vector is equal to the shortest path between the beginning and end of this "chain".== *The order in which you "line up" the constituent vectors does not matter.* - To "visually" subtract two vectors from one another you can **connect the tail of the subtracted vector to the tail of its "subtracee".** This time, the resultant vector is equal to the shortest path between the **subtracted vector's heads**. - NOTE: When subtracting vectors you **can connect the vectors from tip to tail as you would with addition, *if* you switch the direction of the negative vectors**. To find the **resultant vector just connect it from the tail of the first vector to the tip of the last vector**, the same way you do when adding. - A **negative vector** refers to a vector that has been multiplied by -1. It will have the same magnitude as its parent, non negative vector in the **opposite direction**. - To mathematically communicate that you are adding or subtracting vectors **you can use their identifying letter *(with an arrow on top)* in equations just like variables**. For example, $\tiny\vec{A}+\vec{B}=\vec{C}$ or $\tiny\vec{A^2}-\vec{B}=\vec{D}$ - The vector's letters can be replaced with their corresponding magnitudes **when doing calculations**. >[!quote] >##### *Vector math visual* >![[addingandsubtractingvectors.png|312]]![[otherjsalkdf.png|376]]![[Two dimensional motionsdeee.png|167]] >- *The third image shows a vector multiplied by $2$ and $-3$* ##### *Angle vector math* - Algebraically manipulating angle vectors is significantly easier than coordinate vectors. Algebraic operations on the angle $\small\theta$ of these vectors effects their direction and algebraic operations on the associated "value" effects the vector's magnitude. - When adding or subtracting angle vectors we must use **trigonometry** to find the magnitude and angle of the resulting vector. - To find the magnitude or "value" of the resulting vector we can use **either the [[Algebra|Pythagorean theorem]] or the [[Algebra|law of cosines]].** --- ## Vectors and motion - The **net displacement** of an object is equal to the **sum of all the individual displacement vectors over the corresponding period of time**. - We can find the **average acceleration of an object by adding together the initial and final velocity vectors of that object.** - Surprisingly we cannot find the average velocity of an object by adding together vectors. This is because velocity is based on the position of an object at differing points in time and an object's position is a scalar quantity. $\text{Net displacement: }\;\Huge\boxed{\;\large\Delta\vec{x}_{\small net}=\Delta \vec{x}_{\small i}+\Delta \vec{x}_{\small f}}$ $\newline$ $\text{Average acceleration: }\;\Huge\boxed{\;\large \vec{a}_{\small avg}=\Delta\vec{a}_{\small i}+\Delta\vec{a}_{\small f}}$ --- ## Components ##### *The basics* - Vectors **are always made up of two components, the x component and the y component.** The sum of these *component vectors* will always equal the parent vector. Physicist often split up forces/properties **into their individual components like this because it simplifies the math**. - The x and y components of a property/force can be thought of as separate vectors because changes to one component very rarely effect the other component. - Components must **always stay aligned with their corresponding axis**. - Vector components are **usually denoted using a subscript x or y.** For example if the parent vector is $\small\vec{D}$ its component vectors would be: $\small\vec{D_x}\space\space\vec{D_y}$ >[!quote] >##### *Vector components visual* >‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎![[vectorcomponents.png|320]]![[vectorcomponent3.png|341]] ##### *Finding a coordinate vector's components* - Finding the components of a coordinate vector is **very easy**. The **x component of a coordinate vector will have an x value equal to that of its parent vector and a y value equal to zero.** Conversely, the **y component of a coordinate vector will have a y value equal to that of its parent vector and an x value equal to zero.** - To find the parent vector of two component *coordinate vectors* all you have to do is add the two vectors together. ##### *Finding an angle vector's components* - To find the magnitude of an angle vector's the x/y components you must **you can use the $sin$ and $cos$ functions**. The cosine function $\small\vec{V}\cos\theta$ is used to find the magnitude of a vector's x component and the sine $\small\vec{V}\sin\theta$ function is used to find the vector's y component. - If we know only a single component of a vector along with that vector's angle we can find the parent vector if we divide that vector's magnitude by its corresponding trigonometric function. $\newline$ $\text{Components of an angle vector:}$ $\huge\boxed{\color{lightcyan}\;\large\vec{a_x}=\vec {a}\cdot \cos\theta\space\space\space\space\space\vec{a_y}=\vec{a}\cdot \sin\theta\;}$ $\newline$ - If you know the magnitude of a parent vector's components but not of the parent itself **you can use the Pythagorean theorem to find the magnitude of their parent vector**. - Furthermore, using this same information you can find the angle of the parent vector. $\newline$ $\text{Parent vectors magnitude: }\;\;\huge\boxed{\;\color{lightcyan}\normalsize{\huge|}\vec{A}\,{\huge|}=\sqrt{\Big(\vec{A_x}\Big)^2+\Big(\vec{A_y}\Big)^2}\;}$ $\text{Parent vector's angle: }\;\;\Huge\boxed{\;\color{lightcyan}\large\theta=tan^{-1}\left(\frac{\vec{A_y}}{\vec{A_x}}\right)\;}$ $\newline$ >[!quote] >##### *Finding the parent vector visual* >![[componentpythad.png|850]] --- ## Calculator <iframe src="https://app.calconic.com/api/embed/calculator/6488a35028c7470029d3e4db" sandbox="allow-same-origin allow-forms allow-scripts allow-top-navigation allow-popups-to-escape-sandbox allow-popups" title="Calconic_ Calculator" name="Calconic_ Calculator" height="2000" scrolling="no" style="width: 100%; border: 0; outline: none;"></iframe>