Vector Addition

Adding vectors is fundamentally different from adding scalars: direction matters. Three classical graphical laws and one algebraic method cover all cases.

Concept

Triangle Law

To add $\vec{A} + \vec{B}$: place tail of $\vec{B}$ at head of $\vec{A}$. The resultant $\vec{R}$ goes from tail of $\vec{A}$ to head of $\vec{B}$.

Parallelogram Law

Place $\vec{A}$ and $\vec{B}$ tail-to-tail. Complete the parallelogram. The diagonal from the common origin is $\vec{R}$.

For two vectors of magnitudes $A$, $B$ with angle $\theta$ between them: $R = \sqrt{A^2 + B^2 + 2AB\cos\theta}$ $\tan\alpha = \frac{B\sin\theta}{A + B\cos\theta}$ where $\alpha$ is the angle of $\vec{R}$ with $\vec{A}$.

Polygon Law

For $n$ vectors: place them head-to-tail in sequence. The resultant is from the tail of the first to the head of the last.

Algebraic / Component Method

In components: $\vec{R} = \vec{A} + \vec{B} = (A_x + B_x)\hat{i} + (A_y + B_y)\hat{j}.$

Magnitude: $R = \sqrt{R_x^2 + R_y^2}.$

Angle with $x$-axis: $\tan\theta = R_y/R_x$.

Properties

Maximum and Minimum

For two vectors of magnitudes $A$ and $B$:

• Maximum $|\vec{R}| = A + B$ (when parallel, $\theta = 0$).
• Minimum $|\vec{R}| = |A - B|$ (when antiparallel, $\theta = 180°$).

Worked Example

Q: $\vec{A} = 3\hat{i} + 2\hat{j}$ and $\vec{B} = -\hat{i} + 4\hat{j}$. Find (a) $\vec{R} = \vec{A} + \vec{B}$, (b) magnitude and angle with $x$-axis.

Solution: (a) $\vec{R} = (3-1)\hat{i} + (2+4)\hat{j} = 2\hat{i} + 6\hat{j}$.

(b) $|R| = \sqrt{4+36} = \sqrt{40} = 2\sqrt{10} \approx 6.32$.

$\tan\theta = 6/2 = 3 \Rightarrow \theta \approx 71.6°$.

Q (using parallelogram law): Two forces of 5 N and 8 N act at 60° to each other. Find the resultant.

$R = \sqrt{25 + 64 + 2(5)(8)\cos 60°} = \sqrt{89 + 40} = \sqrt{129} \approx 11.36\,\text{N}.$

Angle with 5 N: $\tan\alpha = (8 \sin 60°)/(5 + 8\cos 60°) = 6.93/9 \approx 0.77 \Rightarrow \alpha \approx 37.6°$.

Common Confusions

• "Vectors add by simple arithmetic." — Only when parallel.
• "Triangle law and parallelogram law give different answers." — They give the same answer; just different constructions.
• Adding magnitudes (e.g., $5 + 8 = 13$ N) is only correct for parallel vectors.

Key Takeaways

• Graphical: triangle, parallelogram, polygon laws.
• Algebraic: add components separately.
• $R = \sqrt{A^2 + B^2 + 2AB\cos\theta}$ for two vectors.
• $R_{\max} = A + B$, $R_{\min} = |A - B|$.
• Vector addition is commutative and associative.