<p align="right">Last Update: <font color="#4f81bd">January 07, 2025</font></p> A [[physical quantity]] may be categorized as a [[Scalars|scalar]] quantity or vector quantity. A physical quantity fully described by a number and its unit is called a ==scalar== quantity; examples include mass, temperature, volume, and energy. We use algebraic symbols to represent a scalar quantity. Scalar quantities may be added algebraically: 3 _kg_ + 2 _kg_ = 5 _kg_ A ==vector== is a quantity with ==magnitude== (size) and ==direction==; and is written as a number, unit, and direction. For example, 32 m [S], where the S stands for south. Another way to write this example, is if the coordinate system is set as the frame of reference and south is negative, then 32 m [S] would be -32 m. Examples of vectors include [[Notes Vault/Physics Notes Vault/Kinematics/1D Motion/Position|position]] ($\vec{x}$), [[displacement]] ($\Delta\vec{x}$), [[velocity]] ($\vec{v}$), [[Notes Vault/Physics Notes Vault/Kinematics/1D Motion/Acceleration|Acceleration]] ($\vec a$), [[force]] ($\vec F$), and [[momentum]] ($\vec p$). The small arrow over the symbol for the physical quantity represents that the quantity is a vector. The symbols $\vec {r}$ and _r_ do not represent the same quantity. ### Representing Vectors There are many other ways to represent a vector. We can use a column method $\begin{bmatrix}x\\ y \end{bmatrix}$ , a parentheses method (x, y), or include what are known as vector units (3$\hat{i}$ + 2$\hat{j}$). A geometric representation of a vector is drawn as an arrow. The starting point is the tail and the end of the vector is called the head. The length of the arrow is the magnitude of the vector. The arrow points in the direction of the vector quantity (e.g., position, velocity, acceleration). ![[Screen_Shot_2020-10-14_at_9.33.32_AM.png]] ### Vector Math Combining vectors leads to a **==resultant vector==**. The addition can be done graphically or through trigonometry and the **==Pythagorean theorem==**. To graphically add vectors, the resultant would begin at the beginning of the first vector and end at the end of the last vector. The magnitude of the resultant is found using a ruler and the angle is found with a protractor. Vector subtraction uses the negative as the opposite direction with the same magnitude. Vectors can be multiplied by scalars to result in a vector. For example, a vehicle going twice as fast will double the magnitude and keep the direction of the vector the same. ### Unit Vectors Unit vectors define the x and y directions in space. - Unit vectors have a magnitude of 1 - Unit vectors do no have units. - $\hat{i}$ points in the x direction and $\hat{j}$ points in the y direction. ![[Screen_Shot_2020-10-14_at_8.57.44_AM.png]] ## Vector Components ![[Notes Vault/Physics Notes Vault/Measurements/Vector Components#Definition]] [Read more](Notes%20Vault/Physics%20Notes%20Vault/Measurements/Vector%20Components.md) ### Slide Deck <div style="position: relative; width: 100%; height: 0; padding-top: 56.2500%; padding-bottom: 0; box-shadow: 0 2px 8px 0 rgba(63,69,81,0.16); margin-top: 1.6em; margin-bottom: 0.9em; overflow: hidden; border-radius: 8px; will-change: transform;"> <iframe loading="lazy" style="position: absolute; width: 100%; height: 100%; top: 0; left: 0; border: none; padding: 0;margin: 0;" src="https://www.canva.com/design/DAGYwiaHZb4/Z4PlbBQut1lpOB6N4WJOYQ/view?embed" allowfullscreen="allowfullscreen" allow="fullscreen"> </iframe> </div> <div style="position: relative; width: 100%; height: 0; padding-top: 56.2500%; padding-bottom: 0; box-shadow: 0 2px 8px 0 rgba(63,69,81,0.16); margin-top: 1.6em; margin-bottom: 0.9em; overflow: hidden; border-radius: 8px; will-change: transform;"> <iframe loading="lazy" style="position: absolute; width: 100%; height: 100%; top: 0; left: 0; border: none; padding: 0;margin: 0;" src="https://www.canva.com/design/DAGRo242jW8/Y79kn8i3abWZ8EtFi49XtQ/view?embed" allowfullscreen="allowfullscreen" allow="fullscreen"> </iframe> </div> ### Related Topics --- [[Home|Home]] | [[Measurements]] | [[Scalars]] | [[Vectors]] | [[Tensors]] | [[Spinors]]