University Level Notes: Vectors in 2D and 3D Space

Definition of a Vector: A vector is defined as a quantity characterized by two primary features: its direction and its magnitude.
- Named Example: Velocity is an example of a vector quantity.
Vector Components: In rectangular form, a vector's horizontal and vertical components are expressed as $\langle x, y \rangle$ .
Standard Position: A vector is considered to be in standard position when its tail (starting point) is located at the origin $(0,0)$ .
Representing Vectors as Complex Numbers: A vector with horizontal component $6$ and vertical component $3$ can be represented as the complex number $6 + 3i$ .
Angular Awareness: When dealing with vectors, it is often necessary to keep the radius $r$ positive, which requires careful consideration of the implications placed on the angle.

Vector Addition: Given vectors $u = \langle a, b \rangle$ and $v = \langle c, d \rangle$ , their sum is calculated by adding corresponding components:
- $u + v = \langle a + c, b + d \rangle$
Scalar Multiplication: If $u = \langle a, b \rangle$ and $k$ is a constant (scalar), the scalar product is:
- $k \times u = \langle ka, kb \rangle$
The Norm (Magnitude): The norm or magnitude of a vector is the distance from the point $(x, y)$ to the origin when the vector is in standard position.
- The symbol for magnitude is $|v|$
- Formula: $|v| = \sqrt{x^2 + y^2}$
Unit Vector: A vector with a magnitude exactly equal to $1$ is referred to as a unit vector.
The Zero Vector: Denoted as $0$ , this is a vector of length $0$ where all components are equal to zero.
- The direction of a zero vector is undefined.
- It serves as the additive identity of the additive group of vectors.
Component Form from Two Points: For any two points $A$ and $B$ , the component form of vector $\vec{AB}$ is calculated by subtracting the coordinates of the initial point from the terminal point.
- Example: Find the component form of vector $\vec{CF}$ if $C = (-4, -3)$ and $F = (1, 5)$ .
- Calculation: $\langle 1 - (-4), 5 - (-3) \rangle = \langle 5, 8 \rangle$

Example: Plane Vector Length and Direction: An arrow drawn from terminal point $(3, 5)$ from initial point $(-1, 2)$ represents a plane vector.
- Component form: $\langle 3 - (-1), 5 - 2 \rangle = \langle 4, 3 \rangle$
- Length: $|v| = \sqrt{4^2 + 3^2} = 5$
Example: Force and Components: A vector $\vec{f}$ represents a force of $5\,lb$ exerted at an angle of $\frac{5\pi}{6}$ with the positive x-axis.
- To find the $x$ and $y$ components:
- $x = 5 \times \cos(\frac{5\pi}{6})$
- $y = 5 \times \sin(\frac{5\pi}{6})$
Real-World Scenario: Volleyball: Two players hit a ball at the same time at the net.
- Team 1: Force = $40\,lb$ , Angle = $35^{\circ}$
- Team 2: Force = $30\,lb$ , Angle = $70^{\circ}$
- The winner of the point is determined by the resultant force vector.
Real-World Scenario: Resultant Velocity: A plane is traveling at $200\,m/h$ due northeast. A wind is blowing at $50\,m/h$ from the south.
- Goal: Find the true resultant velocity (the combination of speed and direction).

Scalar Multiples and Direction: Let $v$ be a nonzero vector.
- $3v$ is geometrically related to $v$ by being three times as long in the same direction.
- $-v$ (or $(-1)v$ ) is the same length but in the exact opposite direction.
Definition of Parallel Vectors: Two vectors are called parallel if they are nonzero scalar multiples of each other. This means they share the same or opposite directions.
- Vector $w = (w_1, w_2)$ is a scalar multiple of $u = (u_1, u_2)$ (written as $w = ku$ ) if and only if there exists a nonzero real number $k$ such that $(w_1, w_2) = (ku_1, ku_2)$ .
- Example: Vector $(2, 5)$ is parallel to $(18, 45)$ because $(18, 45) = 9 \times (2, 5)$ .
Vector Equations for Lines in 2D: There is exactly one line parallel to a given line passing through a specific point.
- Let a line pass through point $P(x_0, y_0)$ and be parallel to vector $v(v_1, v_2)$ .
- A point $Q(x, y)$ is on this line if and only if there is a real number $t$ such that:
- Vector Equation: $(x - x_0, y - y_0) = t(v_1, v_2)$
- Parametric Equations: $x = x_0 + tv_1$ and $y = y_0 + tv_2$

Dot Product Definition: The dot product of vectors $u\langle u_1, u_2 \rangle$ and $v\langle v_1, v_2 \rangle$ results in a real number denoted by $u \cdot v$ .
- Formula: $u \cdot v = u_1v_1 + u_2v_2$
Properties of the Dot Product:
- Dot product with self: $w \cdot w = |w|^2 = w_1^2 + w_2^2$
- Commutativity: The dot product is commutative ( $u \cdot v = v \cdot u$ ).
Angle Between Two Vectors Theorem: Suppose $\theta$ is the measure of the angle between two nonzero vectors $u$ and $v$ , where $0 \le \theta \le \pi$ . Then:
- $\cos(\theta) = \frac{u \cdot v}{|u||v|}$
Orthogonality: Two vectors $u$ and $v$ are perpendicular if and only if their dot product is zero ( $u \cdot v = 0$ ). Perpendicular vectors are also called orthogonal.
Geometric Meaning of Negative Dot Product: If $u \cdot v < 0$ , it implies that the angle between the two vectors is obtuse ( $90^{\circ} < \theta \le 180^{\circ}$ ).

3D Coordinates: Points are represented by three coordinates $(x, y, z)$ .
Distance Formula in 3D: The length of segment $AB$ between points $A(x_1, y_1, z_1)$ and $B(x_2, y_2, z_2)$ is:
- $|AB| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$
Equation of a Sphere: A sphere is described by the equation:
- $(x - h)^2 + (y - k)^2 + (z - l)^2 = r^2$
- Example: Verify $x^2 + y^2 + z^2 - 2x + 10z = 10$ is a sphere by completing the square to find center and radius.
Basic 3D Operations:
- Sum: $u + v = (u_1 + v_1, u_2 + v_2, u_3 + v_3)$
- Scalar Multiple: $ku = (ku_1, ku_2, ku_3)$
- Dot Product: $u \cdot v = u_1v_1 + u_2v_2 + u_3v_3$
- Magnitude: $|u| = \sqrt{u_1^2 + u_2^2 + u_3^2}$
Robot Arm Monitoring: A robot arm at origin $O(0, 0, 0)$ moves an object from $(5, 25, 32)$ to $(92, -17, 10)$ .
- Distance traveled is the magnitude of the displacement vector.
- Extension comparison involves comparing the distances from the origin at start and finish.

Cross Product Definition: Given $u = (u_1, u_2, u_3)$ and $v = (v_1, v_2, v_3)$ , the cross product $u \times v$ is a vector:
- $u \times v = (u_2v_3 - v_2u_3, u_3v_1 - v_3u_1, u_1v_2 - v_1u_2)$
Geometric Significance: Unlike the dot product (which yields a scalar), the cross product creates a new vector. This vector is perpendicular to the plane containing the two original vectors.
Properties:
- $u \times v$ is orthogonal to both $u$ and $v$ .
- $u \times v = -(v \times u)$ .

Equations for a Line in 3D: A point $Q(x, y, z)$ lies on line $L$ through point $P(x_0, y_0, z_0)$ parallel to vector $v(v_1, v_2, v_3)$ if there exists a real number $t$ such that:
1. Vector Equation: $PQ = tv$
2. Parametric Equations:
  - $x = tv_1 + x_0$
  - $y = tv_2 + y_0$
  - $z = tv_3 + z0$
Equations for a Plane:
- A plane perpendicular to vector $v(a, b, c)$ is the set of points $(x, y, z)$ satisfying the equation $ax + by + cz = d$ .
- Three noncollinear points are required to uniquely determine a plane.
- Example: To find the equation of a plane through points $P(1, 3, 2)$ , $Q(3, -1, 6)$ , and $R(5, 2, 0)$ , one must find the normal vector using the cross product of $\vec{PQ}$ and $\vec{PR}$ .
Angle Between Planes: The angle between two planes is the angle between their respective normal vectors.