Precalculus Applications of Trigonometry and Vectors Syllabus

How do you add two vectors $\vec{u}=\langle a,b\rangle$ and $\vec{v}=\langle c,d\rangle$?

Add componentwise: $\vec{u}+\vec{v}=\langle a+c,\, b+d\rangle$.

How do you multiply a vector $\vec{v}=\langle a,b\rangle$ by a scalar $k$?

$$k\vec{v}=\langle ka,\, kb\rangle$$ which scales the magnitude by $|k|$ and reverses direction if $k<0$.

What are the standard unit vectors $\vec{i}$ and $\vec{j}$, and how is $\langle a,b\rangle$ written with them?

$\vec{i}=\langle 1,0\rangle$, $\vec{j}=\langle 0,1\rangle$; then $\langle a,b\rangle=a\vec{i}+b\vec{j}$.

How do you find a unit vector in the direction of $\vec{v}$?

Divide the vector by its magnitude: $$\vec{u}=\frac{\vec{v}}{\|\vec{v}\|}$$

How do you write a vector of magnitude $\|\vec{v}\|$ and direction angle $\theta$ in component form?

$$\vec{v}=\langle \|\vec{v}\|\cos\theta,\; \|\vec{v}\|\sin\theta\rangle$$

How do you find the direction angle $\theta$ of a vector $\vec{v}=\langle a,b\rangle$?

$\theta=\tan^{-1}\!\left(\frac{b}{a}\right)$, adjusted to the correct quadrant based on the signs of $a$ and $b$.

What is a resultant vector in applied problems?

The single vector equal to the sum of two or more vectors, representing their combined effect (e.g., total force or net velocity).

In navigation, how are vectors used to find the actual (ground) course of a plane in wind?

Add the plane's air-velocity vector and the wind-velocity vector; the resultant gives the true ground speed (magnitude) and true bearing (direction).

What is the definition of the dot product of $\vec{u}=\langle a_{1},b_{1}\rangle$ and $\vec{v}=\langle a_{2},b_{2}\rangle$?

$$\vec{u}\cdot\vec{v}=a_{1}a_{2}+b_{1}b_{2}$$ (a scalar).

State the geometric (angle) form of the dot product.

$$\vec{u}\cdot\vec{v}=\|\vec{u}\|\,\|\vec{v}\|\cos\theta$$ where $\theta$ is the angle between the vectors.

List three algebraic properties of the dot product.

Commutative: $\vec{u}\cdot\vec{v}=\vec{v}\cdot\vec{u}$; distributive: $\vec{u}\cdot(\vec{v}+\vec{w})=\vec{u}\cdot\vec{v}+\vec{u}\cdot\vec{w}$; and $\vec{v}\cdot\vec{v}=\|\vec{v}\|^{2}$.

How do you find the angle $\theta$ between two vectors using the dot product?

$$\cos\theta=\frac{\vec{u}\cdot\vec{v}}{\|\vec{u}\|\,\|\vec{v}\|}$$

What does the dot product tell you about whether two nonzero vectors are orthogonal (perpendicular)?

They are orthogonal if and only if $\vec{u}\cdot\vec{v}=0$ (since $\cos 90^{\circ}=0$).

For nonzero vectors, what sign of the dot product indicates an acute vs. obtuse angle between them?

Positive dot product $\Rightarrow$ acute angle; negative dot product $\Rightarrow$ obtuse angle.

What is the formula for the scalar projection of $\vec{u}$ onto $\vec{v}$ (the component of $\vec{u}$ along $\vec{v}$)?

$$\text{comp}_{\vec{v}}\vec{u}=\frac{\vec{u}\cdot\vec{v}}{\|\vec{v}\|}$$

What is the vector projection of $\vec{u}$ onto $\vec{v}$?

$$\text{proj}_{\vec{v}}\vec{u}=\left(\frac{\vec{u}\cdot\vec{v}}{\|\vec{v}\|^{2}}\right)\vec{v}$$

How is work computed as a dot product when a constant force $\vec{F}$ acts over displacement $\vec{d}$?

$$W=\vec{F}\cdot\vec{d}=\|\vec{F}\|\,\|\vec{d}\|\cos\theta$$ where $\theta$ is the angle between force and displacement.

How can any vector $\vec{u}$ be decomposed relative to a vector $\vec{v}$?

As the sum of a component parallel to $\vec{v}$ (the projection) and a component orthogonal to $\vec{v}$: $\vec{u}=\text{proj}_{\vec{v}}\vec{u}+\vec{w}$, where $\vec{w}\perp\vec{v}$.

How is a complex number $z=a+bi$ plotted in the complex plane?

As the point $(a,b)$, where the horizontal axis is the real part and the vertical axis is the imaginary part.

What is the modulus (absolute value) of $z=a+bi$?

$$|z|=r=\sqrt{a^{2}+b^{2}}$$

What is the trigonometric (polar) form of a complex number $z=a+bi$?

$$z=r(\cos\theta+i\sin\theta)$$ where $r=\sqrt{a^{2}+b^{2}}$ and $\theta$ is the argument.

How do you convert $z=a+bi$ to polar form (find $r$ and $\theta$)?

$r=\sqrt{a^{2}+b^{2}}$ and $\theta=\tan^{-1}\!\left(\frac{b}{a}\right)$, placing $\theta$ in the quadrant of $(a,b)$.

What is the argument $\theta$ of a complex number?

The angle, measured from the positive real axis, that the segment from the origin to the point $(a,b)$ makes; it satisfies $a=r\cos\theta$, $b=r\sin\theta$.

State the rule for the product of two complex numbers in polar form.

For $z_{1}=r_{1}(\cos\theta_{1}+i\sin\theta_{1})$ and $z_{2}=r_{2}(\cos\theta_{2}+i\sin\theta_{2})$: $$z_{1}z_{2}=r_{1}r_{2}\big[\cos(\theta_{1}+\theta_{2})+i\sin(\theta_{1}+\theta_{2})\big]$$

State the rule for the quotient of two complex numbers in polar form.

$$\frac{z_{1}}{z_{2}}=\frac{r_{1}}{r_{2}}\big[\cos(\theta_{1}-\theta_{2})+i\sin(\theta_{1}-\theta_{2})\big]$$

In words, how do moduli and arguments behave when multiplying and dividing complex numbers?

Multiply/divide the moduli, and add/subtract the arguments respectively.

State DeMoivre's Theorem for powers of a complex number.

For $z=r(\cos\theta+i\sin\theta)$ and integer $n$: $$z^{n}=r^{n}\big(\cos n\theta+i\sin n\theta\big)$$

What is the formula for the $n$ distinct $n$th roots of $z=r(\cos\theta+i\sin\theta)$?

For $k=0,1,\ldots,n-1$: $$z_{k}=r^{1/n}\left[\cos\!\left(\frac{\theta+360^{\circ}k}{n}\right)+i\sin\!\left(\frac{\theta+360^{\circ}k}{n}\right)\right]$$

How many distinct $n$th roots does a nonzero complex number have, and how are they arranged geometrically?

Exactly $n$ distinct roots, equally spaced $\frac{360^{\circ}}{n}$ apart on a circle of radius $r^{1/n}$ centered at the origin.

How do you find the $n$th roots of unity using DeMoivre's Theorem?

Write $1=\cos 0^{\circ}+i\sin 0^{\circ}$ ($r=1,\theta=0$); the roots are $\cos\!\left(\frac{360^{\circ}k}{n}\right)+i\sin\!\left(\frac{360^{\circ}k}{n}\right)$ for $k=0,\ldots,n-1$.