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 covers vector notation, unit vectors, scalar and vector functions, higher-dimensional functions, and introductory vector differentiation and integration.

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Vector Notations

Note* Depending on what book you are in or what level of math/physics you are in, the general symbols for vector are going be $\vec{A}$, $\vec{v}$, $\mathbf{v}$, $v$. These all mean the same thing. You must pay attention to the context of the book, print vs. handwriting and your professor’s choice of notation which may differ from the book. The idea is that you will know what symbol is correct based on the context of the question after some practice.

$$ \vec{A}=\langle x_0,y_0\rangle\in\mathbb{R}^2, \qquad \vec{B}=\langle x_0,y_0,z_0\rangle\in\mathbb{R}^3, \qquad \mathbf{v}=\vec{v}=\langle v_1,v_2,v_3\rangle =v_1\hat{i}+v_2\hat{j}+v_3\hat{k} $$

$$ \hat{i}\equiv\mathbf{i}=\langle1,0,0\rangle, \qquad \hat{j}\equiv\mathbf{j}=\langle0,1,0\rangle, \qquad \hat{k}\equiv\mathbf{k}=\langle0,0,1\rangle \quad [\text{unit vectors}] $$

Unit Vector

Unit Vector: Magnitude or Length (norm) equal to 1. I.e.,

$$ |\mathbf{v}|=\sqrt{v_1^2+v_2^2+v_3^2}=1\text{ unit}. $$

In 2D which is $\mathbb{R}^2$

$$ |\mathbf{v}|=\sqrt{v_1^2+v_2^2}\equiv\|\vec{v}\|\equiv\|\mathbf{v}\|. $$

$$ \langle v_1,v_2,v_3\rangle = v_1\hat{i}+v_2\hat{j}+v_3\hat{k} = v_1\langle1,0,0\rangle + v_2\langle0,1,0\rangle + v_3\langle0,0,1\rangle $$

$$ = \langle v_1,0,0\rangle + \langle0,v_2,0\rangle + \langle0,0,v_3\rangle = \langle v_1,v_2,v_3\rangle. $$

Scalar Vector vs. Vector Function

$$ \vec{v}=\langle1,2,3\rangle, \qquad \mathbf{r}(t)=\langle t,t^2,1-t^3\rangle, \qquad \nabla f\sim\mathbf{F}=\langle x,xy,xyz\rangle $$

Single, Double, Triple and Higher Functions

$$ f(x)=x\in\mathbb{R}^2, \qquad f(x,y)=xy\in\mathbb{R}^3, \qquad f(x,y,z)=xyz\in\mathbb{R}^4 $$

$$ y=x\in 2D, \qquad z=xy\in 3D, \qquad w=xyz\in 4D. $$

Derivatives and Antiderivatives for Vectors

Note* You won’t learn this until Calculus 3 but you are expected to understand it at Physics 1 which is a corequisite for Calculus 1. It is simple which is why you don’t Calculus 3.

$$ \mathbf{r}(t)=\langle x(t),y(t)\rangle \Rightarrow \frac{d\mathbf{r}}{dt} = \frac{d}{dt}\langle x(t),y(t)\rangle = \left\langle\frac{dx}{dt},\frac{dy}{dt}\right\rangle = \langle x'(t),y'(t)\rangle = \langle\dot{x},\dot{y}\rangle. $$

$$ \int \mathbf{r}'(t)\,dt = \int\frac{d\mathbf{r}}{dt}\,dt = \int d\mathbf{r} \Rightarrow \left[ \frac{d\mathbf{r}}{dt} = \frac{d}{dt}\langle x(t),y(t)\rangle \right]dt \Leftrightarrow d\mathbf{r}=d\langle x(t),y(t)\rangle. $$

$$ \Rightarrow \int d\mathbf{r} = \int d\langle x(t),y(t)\rangle = \left\langle \int dx,\int dy \right\rangle = \langle x(t)+c_1,y(t)+c_2\rangle = \langle x(t),y(t)\rangle+\langle c_1,c_2\rangle. $$

Continue Learning with The Ultimate Crash Course

This vector notation 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 vectors, calculus, differential equations, physics, engineering mathematics, and other STEM topics.

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