Vector Representations and Addition (Polar, Unit Vectors, and Addition)

Polar form and vector notation

• Vectors can be expressed in at least two common ways: polar form and Cartesian (component) form. It’s useful to be familiar with both and know how to move between them.
• The polar approach emphasizes magnitude and direction, while Cartesian form emphasizes horizontal and vertical components.
• Notation choice is flexible; you can adopt the convention that best fits the problem or the course.

Unit vectors and directionality

• A unit vector carries direction but has unit magnitude. It serves as a directional indicator.
• If you have a vector \vec{v}, the unit vector in its direction is
  $\hat{v} = \frac{\vec{v}}{|\vec{v}|},$
  provided that $|\vec{v}| \neq 0$.
• Common unit vectors in 2D are $\hat{i}$ (x-direction) and $\hat{j}$ (y-direction). Any vector can be written as a linear combination of these:
  $\vec{v} = v<em>x\hat{i} + v</em>y\hat{j}.$
• In 3D, add $\hat{k}$ for the z-direction: $\vec{v} = v<em>x\hat{i} + v</em>y\hat{j} + v_z\hat{k}.$

Trigonometric foundations for vectors

• Vectors relate to right triangles via the standard trigonometric relationships: cosine, sine, and their projections onto axes.
• If a vector makes an angle $\theta$ with the positive x-axis, its Cartesian components are
  $v<em>x = |\vec{v}| \cos\theta, \quad v</em>y = |\vec{v}| \sin\theta.$
• Magnitude of the vector is
  $|\vec{v}| = \sqrt{v<em>x^2 + v</em>y^2}.$
• These relationships underpin both polar and Cartesian representations.

Polar form of a vector in 2D

• In polar form, a vector with magnitude $r$ and angle $\theta$ is written as
  $\vec{v} = r(\cos\theta \,\hat{i} + \sin\theta \,\hat{j}).$
• Equivalently, its Cartesian coordinates are
  $x = r \cos\theta, \quad y = r \sin\theta.$
• The magnitude-angle pair $(r,\theta)$ is not unique because adding $2\pi$ to the angle represents the same direction; be mindful of angle conventions.

Component form and magnitude-angle conversion

• Given $\vec{v} = v<em>x\hat{i} + v</em>y\hat{j},$ the magnitude and direction are
  $|\vec{v}| = \sqrt{v<em>x^2 + v</em>y^2}, \ \text{and}\ \ \tan\theta = \frac{v<em>y}{v</em>x}.$
• Conversely, if you know the magnitude and angle, you can recover components via
  $v<em>x = |\vec{v}| \cos\theta, \ v</em>y = |\vec{v}| \sin\theta.$

Vector addition: two vectors and multiple vectors

• Graphical view: add vectors head-to-tail to obtain a resultant vector.
• Algebraic view: add components componentwise:
  • For two vectors, $\vec{a} = a<em>x\hat{i} + a</em>y\hat{j}, \vec{b} = b<em>x\hat{i} + b</em>y\hat{j}$
    $\vec{a} + \vec{b} = (a<em>x + b</em>x)\hat{i} + (a<em>y + b</em>y)\hat{j}.$
• For multiple vectors, the resultant is
  $\vec{R} = \sum<em>{k=1}^{n} \vec{v}</em>k = \sum<em>{k=1}^{n} (v</em>{kx}\hat{i} + v<em>{ky}\hat{j}) = \left(\sum</em>{k=1}^{n} v<em>{kx}\right)\hat{i} + \left(\sum</em>{k=1}^{n} v_{ky}\right)\hat{j}.$
• Magnitude of the sum of two vectors (using the angle between them, \theta):
  $|\vec{a} + \vec{b}|^2 = |\vec{a}|^2 + |\vec{b}|^2 + 2|\vec{a}||\vec{b}| \cos\theta,$
  where $\theta$ is the angle between the two vectors.
• If you know components, you can also compute directly: $|\vec{a} + \vec{b}| = \sqrt{(a<em>x + b</em>x)^2 + (a<em>y + b</em>y)^2}.$

Law of cosines and its role in vectors

• Law of cosines (triangle form): for a triangle with sides $a$, $b$, and $c$ opposite angle $\gamma$ between sides $a$ and $b$,
  $c^2 = a^2 + b^2 - 2ab \cos\gamma.$
• In vector addition, if you consider the two sides $|\vec{a}|$ and $|\vec{b}|$ with included angle $\theta$ between them, the resultant side (the magnitude of $\vec{a} + \vec{b}$) satisfies
  $|\vec{a} + \vec{b}|^2 = |\vec{a}|^2 + |\vec{b}|^2 + 2|\vec{a}||\vec{b}| \cos\theta.$
• The angle between vectors also relates to the dot product:
  $\cos\theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}, \ \text{with} \ \vec{a} \cdot \vec{b} = a<em>x b</em>x + a<em>y b</em>y.$
• Triangle area relation (useful in some vector problems):
  $\text{Area} = \frac{1}{2} |\vec{a}| |\vec{b}| \sin\theta.$

Equality of vectors and what it means

• Two vectors are equal if and only if they have the same magnitude and direction:
  $\vec{a} = \vec{b} \iff |\vec{a}| = |\vec{b}| \text{ and } \frac{\vec{a}}{|\vec{a}|} = \frac{\vec{b}}{|\vec{b}|}.$
• In component form, equality also means matching components: $a<em>x = b</em>x \quad\text{and}\quad a<em>y = b</em>y.$
• If two Cartesian representations are equal, their polar representations are also consistent (and vice versa) after converting between forms.

Worked example (illustrative)

• Given vectors in 2D:
  • \vec{a} with magnitude $|\vec{a}| = 5$ and direction $\theta_a = 30^{\circ}$.
  • \vec{b} with magnitude $|\vec{b}| = 4$ and direction $\theta_b = 120^{\circ}$.
• Components:
  • $a_x = 5\cos 30^{\circ} = 5\cdot\frac{\sqrt{3}}{2} \approx 4.33,$
  • $a_y = 5\sin 30^{\circ} = 5\cdot\frac{1}{2} = 2.50.$
  • $b_x = 4\cos 120^{\circ} = 4\left(-\tfrac{1}{2}\right) = -2.00,$
  • $b_y = 4\sin 120^{\circ} = 4\left(\frac{\sqrt{3}}{2}\right) \approx 3.46.$
• Resultant:
  • $R<em>x = a</em>x + b_x \approx 4.33 - 2.00 = 2.33,$
  • $R<em>y = a</em>y + b_y \approx 2.50 + 3.46 = 5.96.$
  • Magnitude: $|\vec{R}| = \sqrt{R<em>x^2 + R</em>y^2} \approx \sqrt{2.33^2 + 5.96^2} \approx 6.40.$
  • Direction: $\theta<em>R = \tan^{-1}\left(\frac{R</em>y}{R_x}\right) \approx \tan^{-1}(2.56) \approx 68.2^{\circ}.$
• This illustrates how polar and Cartesian forms connect and how the magnitude and direction of the sum are obtained from components.

Practical implications and connections

• Graphical intuition (head-to-tail) complements algebraic methods; both are valid and often used together.
• The polar form is convenient for problems with angles and rotations; Cartesian form is straightforward for addition and projections on axes.
• These concepts underpin physics (forces, velocities), engineering (resultants, trajectories), and computer graphics (vector operations).
• Be mindful of notation choices (units, directions, sign conventions) to avoid errors, especially when converting between forms.
• Ethical and practical considerations: accurate vector modeling is essential in design, navigation, and safety-critical systems; misinterpretation of angles or units can lead to incorrect results or hazards.

Quick references: key formulas to memorize

• Polar to Cartesian components:
  $x = r \cos\theta, \quad y = r \sin\theta, \ \vec{v} = x\hat{i} + y\hat{j} = r(\cos\theta \,\hat{i} + \sin\theta \,\hat{j}).$
• Magnitude and direction from components:
  $|\vec{v}| = \sqrt{v<em>x^2 + v</em>y^2}, \qquad \theta = \tan^{-1}\left(\frac{v<em>y}{v</em>x}\right).$
• Component form of a vector: $\vec{v} = v<em>x\hat{i} + v</em>y\hat{j}.$
• Equality of vectors (component form): $\vec{a} = \vec{b} \iff a<em>x = b</em>x \text{ and } a<em>y = b</em>y.$
• Sum of multiple vectors: $\vec{R} = \sum<em>{k=1}^{n} \vec{v}</em>k = (\sum<em>{k=1}^{n} v</em>{kx})\hat{i} + (\sum<em>{k=1}^{n} v</em>{ky})\hat{j}.$
• Magnitude of sum (two vectors) with angle between them:
  $|\vec{a} + \vec{b}|^2 = |\vec{a}|^2 + |\vec{b}|^2 + 2|\vec{a}||\vec{b}| \cos\theta.$
• Dot product and angle between vectors:
  $\vec{a} \cdot \vec{b} = a<em>x b</em>x + a<em>y b</em>y, \quad \cos\theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}.$