The Ultimate Crash Course for STEM Majors

This is nothing more than a free preview/sample of The Ultimate Crash Course for STEM Majors. This lesson introduces plotting vectors, repositioning vectors at the origin, vector components, and the basic method used to find a resultant vector.

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Plotting Vectors and Summation

There are many ways to approach identifying what is often called “The Resultant Vector.” I will show you the fool proof simplified way that works every time.

You may see vectors plotted as shown

The Resultant Vector is from the tail to tip

Vector layout: multiple vectors may initially appear connected head-to-tail across an $x,y$ coordinate plane.

The individual vectors can be represented as $\,\vec{A},\vec{B},\vec{C}\,$ with the resultant extending from the original tail to the final tip.

The same vectors can be viewed with the resultant vector drawn from the initial origin to the final endpoint.

$$\vec{R}=\vec{A}+\vec{B}+\vec{C}$$

When you are looking at vectors, the best way to do this is to reposition all the vectors at the origin. You can use sine and cosine without guessing the angles or using silly angle-angle-side B.S. from trig.

After repositioning, all vectors share a common tail at the origin.

$$\vec{A},\qquad\vec{B},\qquad\vec{C},\qquad\vec{R}$$

Each vector is then measured from the positive $x$-axis.

When each vector is shifted to have the tail at the origin, you can find all the $x,y$ components using the angle from the origin with sine for $y$ and cosine for $x$. No critical thinking is involved in finding the resultant when approaching it this way.

$$\vec{A},\vec{B},\vec{C},\vec{R}$$

To find the Resultant vector which will be considered the total force or work or whatever chapter you’re in, the math and arithmetic is nearly identical—that is, the names just change.

You can now take the value that represents the function and break it up into its components and then sum together to find the resultant.

$$\vec{A}=\langle |\vec{A}|\cos\alpha,|\vec{A}|\sin\alpha\rangle,\qquad \vec{B}=\langle |\vec{B}|\cos\beta,|\vec{B}|\sin\beta\rangle,\qquad \vec{C}=\langle |\vec{C}|\cos\gamma,|\vec{C}|\sin\gamma\rangle.$$

Continue Learning with The Ultimate Crash Course

This vector lesson is nothing more than a free preview/sample of The Ultimate Crash Course for STEM Majors. Continue with the series for additional lessons covering vector components, resultant vectors, trigonometry, forces, work, calculus, physics, engineering mathematics, and other STEM topics.

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