Study Notes: Vectors, Matrices, and Vector-Valued Functions in Calculus (Math 200)

Vectors and Geometry in Rn

Geometric Figures and Distance

Understanding geometric figures and distances in Rn is foundational for multivariable calculus. These concepts help describe shapes, locations, and relationships between points in space.

• Geometric Figures: Equations such as $y = x^2$ describe curves (parabolas) in the plane R2.

• Graphs of Objects: Examples include $f(x, y) = x^2$, $y^2 + x^2 = 9$ (a circle of radius 3), and $x^2 + y^2 = 9$.

• Distance Formula: The distance between points $(x_1, y_1)$ and $(x_2, y_2)$ in R2 is given by: $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

• Comparing Points: The position of points can be compared using the axes; for example, which point is 'above' another depends on the $y$-coordinate.

• Example: The distance between $P(1,2,3)$ and $Q(0,0,-1)$ in R3 is $\sqrt{(1-0)^2 + (2-0)^2 + (3-(-1))^2} = \sqrt{1 + 4 + 16} = \sqrt{21}$.

Functions of Several Variables and Level Curves

Revenue, BMI, and Level Curves

Functions of several variables model real-world phenomena such as revenue and body mass index (BMI). Level curves help visualize these functions.

• Revenue Function: For a cable company, revenue $R$ can be modeled as $R = 100m + 5n$, where $m$ is the number of subscribers and $n$ is the number of special feature movies.

• BMI Table: BMI is calculated as $\text{BMI} = \frac{703 \times \text{weight (lb)}}{\text{height (in)}^2}$.

• Level Curves: For a function $f(x, y)$, level curves are the sets where $f(x, y) = k$ for constant $k$. These curves help visualize the function's behavior.

• Example: For $f(x, y) = x^2 + y^2$, level curves are circles centered at the origin with radius $\sqrt{k}$.

Weight (lb)	Height (in)	BMI
120	60	23.4
140	60	27.3
160	60	31.2
180	60	35.2
200	60	39.1

Vectors and Vector Operations

Vector Addition, Subtraction, and Unit Vectors

Vectors are quantities with both magnitude and direction. Operations on vectors are essential for describing motion and geometry in space.

• Vector Addition: $\vec{A} + \vec{B}$ is performed component-wise.

• Vector Subtraction: $\vec{A} - \vec{B}$ is also component-wise.

• Unit Vector: A vector of length 1 in the direction of $\vec{v}$ is $\frac{\vec{v}}{|\vec{v}|}$.

• Example: For $\vec{a} = \langle 2, -1, -2 \rangle$, the unit vector is $\frac{1}{\sqrt{2^2 + (-1)^2 + (-2)^2}} \langle 2, -1, -2 \rangle$.

Dot Product and Orthogonality

Dot Product and Angle Between Vectors

The dot product measures how much two vectors point in the same direction. It is used to determine orthogonality and the angle between vectors.

• Dot Product Formula: For $\vec{a} = \langle a_1, a_2, a_3 \rangle$ and $\vec{b} = \langle b_1, b_2, b_3 \rangle$: $\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3$

• Angle Between Vectors: $\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta$

• Orthogonality: Vectors are orthogonal if $\vec{a} \cdot \vec{b} = 0$.

• Example: If $\vec{a} = \langle 1, 2, 3 \rangle$ and $\vec{b} = \langle 4, -5, 6 \rangle$, then $\vec{a} \cdot \vec{b} = 1\times4 + 2\times(-5) + 3\times6 = 4 - 10 + 18 = 12$.

Matrices and Determinants

Determinants of 2x2 and 3x3 Matrices

Matrices are used to represent systems of equations and transformations. The determinant is a scalar value that can indicate invertibility and area/volume scaling.

• 2x2 Determinant: For $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$, $\det = ad - bc$.

• 3x3 Determinant: For $\begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}$: $a(ei - fh) - b(di - fg) + c(dh - eg)$

• Application: Determinants are used in calculating areas, volumes, and solving systems of equations.

Cross Product and Applications

Cross Product and Area Calculations

The cross product of two vectors in R3 produces a vector perpendicular to both. It is used to compute areas and describe planes.

• Cross Product Formula: For $\vec{a} = \langle a_1, a_2, a_3 \rangle$, $\vec{b} = \langle b_1, b_2, b_3 \rangle$: $\vec{a} \times \vec{b} = \langle a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1 \rangle$

• Area of Parallelogram: $|\vec{a} \times \vec{b}|$

• Area of Triangle: $\frac{1}{2}|\vec{a} \times \vec{b}|$

• Example: For $\vec{a} = \langle 1, 0, 0 \rangle$, $\vec{b} = \langle 0, 1, 0 \rangle$, $\vec{a} \times \vec{b} = \langle 0, 0, 1 \rangle$.

Equations of Planes

Planes in R3

Planes are defined by points and normal vectors. The equation of a plane is fundamental in multivariable calculus and geometry.

• Equation of a Plane: Given point $(x_0, y_0, z_0)$ and normal vector $\vec{n} = \langle a, b, c \rangle$: $a(x - x_0) + b(y - y_0) + c(z - z_0) = 0$

• Standard Form: $ax + by + cz = d$

• Example: The plane through $(1,2,3)$ with normal $\langle 4,5,6 \rangle$ is $4(x-1) + 5(y-2) + 6(z-3) = 0$.

Vector-Valued Functions

Definitions and Properties

Vector-valued functions describe curves and motion in space. Their calculus involves limits, derivatives, and integrals.

• Limit of Vector-Valued Function: $\lim_{t \to a} \vec{r}(t) = \langle \lim_{t \to a} f(t), \lim_{t \to a} g(t), \lim_{t \to a} h(t) \rangle$

• Derivative: $\vec{r}'(t) = \langle f'(t), g'(t), h'(t) \rangle$

• Tangent Vector: $\vec{T}(t) = \frac{\vec{r}'(t)}{|\vec{r}'(t)|}$

• Indefinite Integral: $\int \vec{r}(t) dt = \langle \int f(t) dt, \int g(t) dt, \int h(t) dt \rangle$

• Example: For $\vec{r}(t) = \langle t, t^2, t^3 \rangle$, $\vec{r}'(t) = \langle 1, 2t, 3t^2 \rangle$.

Quantity	Formula
Position	$\vec{r}(t)$
Velocity	$\vec{v}(t) = \vec{r}'(t)$
Acceleration	$\vec{a}(t) = \vec{v}'(t)$
Speed	$|\vec{v}(t)|$ (scalar)

Applications and Problem Solving

Examples and Exercises

Applying these concepts to solve problems is essential for mastering calculus. Problems include finding distances, areas, equations of planes, and analyzing motion.

• Finding the Area of a Triangle: Use the cross product of two sides.

• Finding the Equation of a Plane: Use three points to determine the normal vector and apply the plane equation.

• Analyzing Motion: Use vector-valued functions to describe position, velocity, and acceleration.

• Example: Given $\vec{r}(t) = \langle t, t^2, t^3 \rangle$, find velocity and acceleration: $\vec{v}(t) = \langle 1, 2t, 3t^2 \rangle$, $\vec{a}(t) = \langle 0, 2, 6t \rangle$.

Summary Table: Key Formulas

Concept	Formula
Distance in Rn	$\sqrt{\sum_{i=1}^n (x_i - y_i)^2}$
Dot Product	$\vec{a} \cdot \vec{b} = \sum a_i b_i$
Cross Product	$\vec{a} \times \vec{b}$ (see above)
Equation of Plane	$a(x - x_0) + b(y - y_0) + c(z - z_0) = 0$
Derivative of Vector Function	$\vec{r}'(t) = \langle f'(t), g'(t), h'(t) \rangle$
Indefinite Integral	$\int \vec{r}(t) dt = \langle \int f(t) dt, \int g(t) dt, \int h(t) dt \rangle$

Additional info: These notes cover foundational topics in multivariable calculus, including vectors, matrices, vector-valued functions, and their applications to geometry and motion. The exercises and examples are typical for a college-level Calculus II or III course.