Vectors, Coordinate Systems, Complex Numbers, and Analytic Geometry: Precalculus Study Notes

Vectors and Their Properties

Magnitude of a Vector

The magnitude (or length) of a vector v = <v1, v2, v3> in three dimensions is calculated using the formula:

• Formula: $|\vec{v}| = \sqrt{v_1^2 + v_2^2 + v_3^2}$

• Example: For $\vec{a} = \langle 9, 7, 5 \rangle$, $|\vec{a}| = \sqrt{9^2 + 7^2 + 5^2} = \sqrt{81 + 49 + 25} = \sqrt{155}$

Unit Vector in the Direction of a Given Vector

A unit vector has a magnitude of 1 and points in the same direction as the original vector. To find the unit vector in the direction of v:

• Formula: $\text{Unit vector} = \frac{\vec{v}}{|\vec{v}|}$

• Example: For $\vec{c} = \langle -3, 7, 2 \rangle$, $|\vec{c}| = \sqrt{(-3)^2 + 7^2 + 2^2} = \sqrt{9 + 49 + 4} = \sqrt{62}$, so the unit vector is $\frac{1}{\sqrt{62}}\langle -3, 7, 2 \rangle$

Projection of One Vector onto Another

The projection of vector a onto vector c is the component of a in the direction of c:

• Formula: $\text{proj}_{\vec{c}} \vec{a} = \frac{\vec{a} \cdot \vec{c}}{|\vec{c}|^2} \vec{c}$

• Dot Product: $\vec{a} \cdot \vec{c} = a_1c_1 + a_2c_2 + a_3c_3$

• Example: $\vec{a} = \langle 9, 7, 5 \rangle$, $\vec{c} = \langle -3, 7, 2 \rangle$

Angle Between Two Vectors

The angle $\theta$ between vectors a and c is found using the dot product:

• Formula: $\cos \theta = \frac{\vec{a} \cdot \vec{c}}{|\vec{a}| |\vec{c}|}$

• Example: Calculate $\vec{a} \cdot \vec{c}$, $|\vec{a}|$, $|\vec{c}|$, then $\theta = \arccos(\text{value})$

Coordinate Systems: Rectangular and Polar

Converting Rectangular Coordinates to Polar

To convert $(x, y)$ to polar $(r, \theta)$:

• Formulas:

  • $r = \sqrt{x^2 + y^2}$

  • $\theta = \arctan\left(\frac{y}{x}\right)$

• Example: $(6, 7)$ to polar: $r = \sqrt{6^2 + 7^2} = \sqrt{85}$, $\theta = \arctan(7/6)$

Converting Polar Coordinates to Rectangular

To convert $(r, \theta)$ to rectangular $(x, y)$:

• Formulas:

  • $x = r \cos \theta$

  • $y = r \sin \theta$

• Example: $r = 7$, $\theta = 5\pi/6$: $x = 7 \cos(5\pi/6)$, $y = 7 \sin(5\pi/6)$

Complex Numbers and Their Roots

Complex Numbers in Rectangular and Polar Form

A complex number $z = a + bi$ can be written in polar form as $z = r(\cos \theta + i \sin \theta)$, where:

• $r = \sqrt{a^2 + b^2}$

• $\theta = \arctan\left(\frac{b}{a}\right)$

Finding Roots of Complex Numbers

The nth roots of a complex number $z = r(\cos \theta + i \sin \theta)$ are:

• Formula: $z_k = \sqrt[n]{r}\left[\cos\left(\frac{\theta + 2\pi k}{n}\right) + i \sin\left(\frac{\theta + 2\pi k}{n}\right)\right]$, for $k = 0, 1, ..., n-1$

• Example: Fourth roots of $2 - 2\sqrt{3}i$

Analytic Geometry: Conic Sections

Classification of Conic Sections

Conic sections include hyperbolas, parabolas, and ellipses. Their equations and properties are:

• Hyperbola: $\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$ or $\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1$

• Parabola: $(y-k)^2 = 4p(x-h)$ or $(x-h)^2 = 4p(y-k)$

• Ellipse: $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$

Each conic has a center, vertices, foci, and axes that can be determined from its equation.

Solving Equations and Identifying Conics

• Given a general quadratic equation, classify the conic by comparing coefficients and completing the square if necessary.

• Find the center, vertices, and foci using standard forms.

Trigonometric Identities and Applications

Expressing Trigonometric Functions in Terms of Others

Trigonometric identities allow you to express one function in terms of others. For example:

• Double Angle Formula: $\cos(2x) = \cos^2(x) - \sin^2(x)$

• Expressing $\cos(3x)$: $\cos(3x) = 4\cos^3(x) - 3\cos(x)$

Graphing Trigonometric Functions

• To graph $f(x) = 4\sin(x)$, plot the sine curve and scale the amplitude by 4.

Summary Table: Conic Sections

Conic Type	Standard Equation	Key Features
Hyperbola	$\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$	Center, vertices, foci, asymptotes
Parabola	$(y-k)^2 = 4p(x-h)$	Vertex, focus, directrix
Ellipse	$\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$	Center, vertices, foci

Additional info:

• Some answers and solutions are provided in the file, which reinforce the above concepts.

• Topics covered are directly relevant to Precalculus chapters on vectors, coordinate systems, complex numbers, conic sections, and trigonometric identities.