Chapter 3: Vectors and Coordinate Systems

1. What is a vector and why do we treat it differently?

A scalar (like mass or temperature) only has a size. A vector has size + direction—so where it points matters. Whenever we add or subtract several vectors, the single vector that replaces them is called the resultant. Graphically we find it by placing vectors tip-to-tail; draw one arrow’s tail at the previous arrow’s tip, then join the first tail to the last tip for the resultant direction .

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2. Graphical vs. analytical methods

Tip-to-tail drawings are great for visualising but impractical for calculations—a small protractor error gives a big numeric error. Instead, physicists switch to an analytical (component-based) method inside a Cartesian $x$–$y$ (and sometimes $z$) grid .

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3. Breaking a vector into components

Place the vector so its tail sits at the origin. Using basic trigonometry:

$$A_x = A\cos\theta, \qquad A_y = A\sin\theta$$

where $\theta$ is measured from the $x$-axis. Think of $A_x$ and $A_y$ as the “shadows” the arrow casts on the axes .

Sign hints: • Quadrant tells you plus/minus for each component. • Your calculator’s $\tan^{-1}$ only returns angles between $-90^\circ$ and $+90^\circ$; always check the signs afterward .

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4. Reassembling components

Need the size and overall direction back? Reverse the process:

$$A = \sqrt{A_x^{\,2}+A_y^{\,2}}, \qquad \theta = \tan^{-1}\!\left(\frac{A_y}{A_x}\right)$$

Again, adjust $\theta$ into the correct quadrant .

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5. Unit-vector notation

An even cleaner way to write vectors is with unit vectors $\hat{\mathbf ı},\hat{\mathbf ȷ},\hat{\mathbf k}$ that each point one unit along an axis:

$$\mathbf A = A_x\hat{\mathbf ı} + A_y\hat{\mathbf ȷ}\; (\text{+}\,A_z\hat{\mathbf k})$$

The little “hats” just tell direction; the numbers in front still hold the size .

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6. Adding or subtracting vectors with components

1. Decompose every vector into $x$ and $y$ pieces.
2. Add or subtract all $x$-components to get $C_x$; do the same for $y$ to get $C_y$.
3. Re-assemble $C_x$ and $C_y$ into your final resultant $\mathbf C$ just like in Section 4 .

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7. Quick self-check

Problem: A delivery drone flies $25 \text{m}$ at $35^\circ$ above + $x$ (east) and then $18 \text{m}$ due north. a) What are the components of each leg? b) Find the drone’s total displacement (magnitude & direction).

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Key take-aways

• Vectors differ from scalars because direction counts.
• Use components (and unit-vector notation) to turn messy 2-D vector problems into two neat 1-D problems.
• Always track signs and quadrants.
• Rebuild the magnitude and angle only after finishing all component math.