Two-Dimensional Vectors, Rectangular and Polar Forms, and Engineering Applications

Visual Representation:
- Vectors are visually mapped on a standard XY Cartesian coordinate plane as directed line segments (arrows).
- Every vector features a distinct starting point (origin), an ending point (destination point), and an arrowhead indicating the direction of travel.
- Path tracking from an origin point $O$ to a terminal destination point $P$ is explicitly designated as vector $\vec{OP}$ or simply vector $\mathbf{P}$ .
Textual Notation Standards in Technical Literature:
- Standard top arrow notation: $\vec{P}$
- Hat notation: $\hat{P}$
- Bold formatting: $\mathbf{P}$
- Terminal coordinates inside brackets: $P = [x, y]$ (for example, if point $P$ is located at $3$ on the x-axis and $7$ on the y-axis, it is denoted as $P = [3, 7]$ ).
Fundamental Physical Attributes:
- Magnitude: Defines the total physical scalar length of the vector line segment.
- Direction: Defined by an angle calculated relative to a baseline reference orientation (typically established as the positive x-axis).
- Linear Component Shifts: Measured as localized scalar magnitudes defining exact horizontal translation along the x-axis and vertical translation along the y-axis.

Definition of a Unit Vector:
- A unit vector is defined as a dimensionless vector possessing a magnitude of exactly $1$ that points exclusively along a single coordinate axis.
Standard Coordinate Axis Unit Vectors:
- $\hat{i}$ (i-hat): Unit vector tracking exclusively along the horizontal x-axis with a magnitude of $1$ .
- $\hat{j}$ (j-hat): Unit vector tracking exclusively along the vertical y-axis with a magnitude of $1$ .
- $\hat{k}$ (k-hat): Unit vector tracking exclusively along the orthogonal z-axis with a magnitude of $1$ in three-dimensional coordinate space.
Rectangular Form Construction:
- Vectors are written by multiplying individual linear component magnitudes by their corresponding unit directional vectors.
- General Cartesian equation format: $\mathbf{P} = A_x \hat{i} + A_y \hat{j}$
- Concrete Example: A vector tracking a horizontal length of $4$ along the x-axis and a vertical length of $4$ along the y-axis is expressed as $\mathbf{P} = 4\hat{i} + 4\hat{j}$ .
- Role of the Addition Symbol: The plus sign ( $+$ ) functions strictly as a structural delimiter separating orthogonal x and y component directions; component lengths cannot be directly added into a single scalar sum.

Right-Triangle Geometric Equivalence:
- Two-dimensional vectors map directly to right-angled triangles.
- The horizontal component ( $x$ ) corresponds to the adjacent triangle side.
- The vertical component ( $y$ ) corresponds to the opposite triangle side.
- Vector magnitude ( $P$ ) corresponds to the hypotenuse.
- Vector magnitudes satisfy the Pythagorean theorem: $P^2 = x^2 + y^2$ .
Polar Form Definition:
- Represents a vector state using an overall resultant magnitude ( $P$ ) alongside a directional heading angle ( $\theta$ ).
Conversion Equations:
- Polar to Rectangular Form:
- Horizontal component: $x = P \times \cos(\theta)$
- Vertical component: $y = P \times \sin(\theta)$
- Rectangular to Polar Form:
- Vector magnitude: $P = \sqrt{x^2 + y^2}$
- Direction angle: $\theta = \tan^{-1}\left(\frac{y}{x}\right)$
Quadrant Resolution in Computing:
- Programming functions such as atan2 in MATLAB automatically determine accurate quadrant placement across a full $360^\circ$ orientation ( $2\pi\,\text{rad}$ ) based on signed component inputs.
Physical Modeling Applications:
- Vector magnitudes and directions are used to model real-world physical phenomena including velocity, relative velocity, external forces, acceleration, friction mechanics, and equilibrium systems governed by Newton's first law.

System Configuration:
- Single-link robotic arm system modeled as a standard two-dimensional vector.
- Link length magnitude: $L = 1.5\,\text{ft}$
- Position orientation: Direction angle specified as $1.5\,\text{rad}$ ( $\theta = -30^\circ$ relative to the horizontal baseline).
Step-by-Step Component Resolution:
- Horizontal component ( $x$ ):
- $x = L \times \cos(30^\circ)$
- $x = 1.5 \times \cos(30^\circ)$
- $x \approx 1.299\,\text{ft}$ (points along the positive x-axis)
- Vertical component ( $y$ ):
- Magnitude calculation: $1.5 \times \sin(30^\circ) = 0.75\,\text{ft}$
- Physical direction: Oriented downward into negative y-territory.
- $y = -0.75\,\text{ft}$
Output Form Expressions:
- Rectangular Form: $\mathbf{P} = 1.3\hat{i} - 0.75\hat{j}$
- Polar Form: Overall magnitude of $1.5\,\text{ft}$ at an explicit orientation angle of $-30^\circ$ relative to the horizontal reference axis.

Circuit Network Parameters:
- Series electrical circuit containing a resistor ( $R$ ) connected to an inductor ( $L$ ).
- Phase relationship: Inductor voltage ( $V_L$ ) leads current, creating a fixed $90^\circ$ orthogonal phase shift relative to resistor voltage ( $V_R$ ).
- Voltage magnitudes: Resistor voltage $V_R = 10\,\text{V}$ , Inductor voltage $V_L = 15\,\text{V}$ .
Component Vector Equations (Rectangular Form):
- Resistor voltage vector: $V_R = 10\hat{i} + 0\hat{j}\,\text{V}$ (aligned horizontally on the positive x-axis)
- Inductor voltage vector: $V_L = 0\hat{i} + 15\hat{j}\,\text{V}$ (aligned vertically on the positive y-axis)
- Total combined voltage vector ( $V_{\text{total}}$ ): $V_{\text{total}} = 10\hat{i} + 15\hat{j}\,\text{V}$
- Scalar Non-Equivalence: Alternating current (AC) circuit voltages cannot be summed directly as simple scalars ( $10\,\text{V} + 15\,\text{V} \neq 25\,\text{V}$ ).
Conversion to Polar Form:
- Combined Voltage Magnitude ( $V$ ):
- $V = \sqrt{10^2 + 15^2}$
- $V = \sqrt{100 + 225}$
- $V = \sqrt{325}$
- $V \approx 18.03\,\text{V}$
- Circuit Phase Angle ( $\theta$ ):
- $\theta = \tan^{-1}\left(\frac{15}{10}\right)$
- $\theta = \tan^{-1}(1.5)$
- $\theta \approx 56.3^\circ$
Final Polar Expression:
- Effective circuit voltage magnitude of $18.03\,\text{V}$ at a directional phase angle of $56.3^\circ$ .