Vector Addition

Vectors obey rules of geometry, not arithmetic. To add forces, velocities or displacements you must respect direction. The triangle and parallelogram laws give magnitude and direction in closed form, while component resolution turns any sum into easy algebra.

Concept

Triangle law: Place the tail of $\vec b$ at the tip of $\vec a$. The sum $\vec a + \vec b$ runs from the start to the final tip.

Parallelogram law: Place tails of $\vec a, \vec b$ together; the diagonal of the parallelogram from the common point is $\vec a + \vec b$.

Component form: $\vec a = a_x\hat i + a_y\hat j$, $\vec b = b_x\hat i + b_y\hat j$, so $\vec a + \vec b = (a_x + b_x)\hat i + (a_y + b_y)\hat j$.

If $\theta$ is the angle between $\vec a$ and $\vec b$, $|\vec a + \vec b| = \sqrt{a^{2} + b^{2} + 2 a b \cos\theta}.$

The resultant makes angle $\alpha$ with $\vec a$ where $\tan\alpha = \dfrac{b\sin\theta}{a + b\cos\theta}$.

Derivation

Drop a perpendicular from the tip of $\vec b$ onto the line of $\vec a$ extended. The component along $\vec a$ is $b\cos\theta$; perpendicular is $b\sin\theta$. By Pythagoras on the right triangle formed by $a + b\cos\theta$ and $b\sin\theta$: $R^{2} = (a + b\cos\theta)^{2} + (b\sin\theta)^{2} = a^{2} + 2ab\cos\theta + b^{2}.$ The direction follows from $\tan\alpha = b\sin\theta/(a + b\cos\theta)$.

Extremes: $\theta = 0$ gives $R = a + b$ (maximum); $\theta = \pi$ gives $R = |a - b|$ (minimum); $\theta = \pi/2$ gives $R = \sqrt{a^{2}+b^{2}}$.

JEE Worked Example

Q. Two forces $\vec F_1$ and $\vec F_2$ act at a point with magnitudes $5\,\text{N}$ and $12\,\text{N}$ inclined at $90^\circ$. Find the magnitude and direction of the resultant.

Solution. $R = \sqrt{5^{2} + 12^{2}} = 13\,\text{N}$. Angle with $\vec F_1$: $\tan\alpha = 12/5$, so $\alpha = \tan^{-1}(12/5) \approx 67.4^\circ$.

Traps

• Adding magnitudes scalarly when directions differ.
• Forgetting that $R_{\min} = |a - b|$ may be zero only if $a = b$ and $\theta = \pi$.
• Using the parallelogram formula with the supplementary angle (always use the angle between when both tails coincide).
• Mixing degrees and radians in a calculator.

Key Takeaways

• Resolve into perpendicular components for any non-trivial sum.
• Memorise $R^{2} = a^{2} + b^{2} + 2ab\cos\theta$ and its extremes.
• Direction comes from $\tan\alpha = b\sin\theta/(a + b\cos\theta)$.