<p align="right">Last Update: <font color="#4f81bd">December 08, 2024</font></p> ## Definition Components are parallel to the coordinate axes. They are labeled with a subscript. For example, the component $v_y$ is the $y$-component of a vector. ## Importance Vector components are extremely important for physics students. They are typically drawn with ==**dashed lines**==, and a subscript states the x or y axis. The dash and subscripts help to distinguish between vectors and vector components. ## Right Triangle Trigonometry The vector components of a [[Vectors|vector]] are best understood applying right triangle trigonometry. The angle next to the tail end is very important to know or is something to be determined in a physics problem. [](https://docs.google.com/drawings/u/0/d/sYGj37Jnc9lZ-TkGi1-u2Sw/image?w=173&h=113&rev=1&ac=1&parent=1yAPuP9uJrLKqzLf2uXutXkkMDGTeO0G5cvr7W8rpUUE) Let’s return to the vector $\vec r$. It has a horizontal vector component of 3 and a vertical component of 2. We represent the horizontal vector as $r_x$, where the subscript x represents the x-axis. We represent the vertical vector as $r_y$, where the subscript y represents the y-axis. [](https://docs.google.com/drawings/u/0/d/sdjtM2hg_uKMt8EUZaBy_7g/image?w=173&h=107&rev=1&ac=1&parent=1yAPuP9uJrLKqzLf2uXutXkkMDGTeO0G5cvr7W8rpUUE) Vectors can be moved parallel to themselves, therefore, slide the vertical component to the right and make a complete triangle. The tip of the $r_x$ is matched up to the tail of the $r_y$ component. This is called the ==**tip to tail method**==. We have now created a right triangle and may use the information to find the length of the r vector -- the hypotenuse of the two vector components. If we are not given an angle of the vector r, we can apply the ==**Pythagorean theorem**== and solve for the length. $\vec{r}=\sqrt{r_x^2 + r_y^2}$ Solving for the vectors returns a value of 3.6. Of course, we are being sloppy with significant figures. We’d really want the vector components to be 2.0 and 3.0. What happens if we are given an angle and only the length of one of the components? Can we still solve for the ==**resultant vector**== (the hypotenuse)? ### 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/DAGYwgMZ6Zw/_NfC545opxBL-EzH2hF8Ww/view?embed" allowfullscreen="allowfullscreen" allow="fullscreen"> </iframe> </div> ### Related Topics --- [[Home|Home]] | [[Measurements]] | [[Scalars]] | [[Vectors]] | [[Tensors]] | [[Spinors]]