# Vector Addition — Triangle Law, Parallelogram Law, Examples ## TL;DR Vector addition combines two vectors into a single resultant that captures their joint effect. This article covers the three methods — the triangle law, the parallelogram law, and the component method — the resultant-magnitude formula ∣R⃗∣=P²+Q²+2PQcosθ, and the mistakes students make most. ## What Is Vector Addition? **Vector addition** is the operation that combines two or more vectors into one **resultant vector** — a single vector that produces the same effect as the originals acting together. Because vectors carry direction, you cannot add their magnitudes the way you add ordinary numbers; you have to account for the angle between them. There are three standard ways to do it, and they all agree on the answer: the **triangle law**, the **parallelogram law**, and the **component method**. The first two are geometric pictures; the third is the algebra you'll use most once vectors are in component form. ## How Do You Add Two Vectors With The Triangle Law? The **triangle law of vector addition** states that if two vectors are represented, in magnitude and direction, by two sides of a triangle taken **in order**, then their resultant is the third side taken in the **opposite order**. In practice you place the **tail of the second vector at the head of the first** (the head-to-tail method), and the resultant runs from the tail of the first to the head of the second. When the two vectors \( \vec{P} \) and \( \vec{Q} \) meet at angle \( \theta \), the resultant's magnitude and direction are: ∣R⃗∣=P²+Q²+2PQcosθ ϕ=tan⁻¹\left(\frac{Qsinθ}{P+Qcosθ}\right) This magnitude formula comes from dropping a perpendicular from the head of \( \vec{Q} \) to the line of \( \vec{P} \) and applying the Pythagorean theorem to the extended triangle — the 2PQcosθ term is what survives from squaring the projected pieces. **Variable glossary:** P=∣P⃗∣, Q=∣Q⃗∣ are the magnitudes; θ is the angle between the vectors when drawn from a common tail; ∣R⃗∣ is the resultant magnitude; ϕ is the angle the resultant makes with \( \vec{P} \). ## How Is The Parallelogram Law Different? The **parallelogram law of vector addition** uses the _same_ two vectors drawn from a **common tail** rather than head to tail. If \( \vec{P} \) and \( \vec{Q} \) form two adjacent sides of a parallelogram, the resultant is the **diagonal** drawn from their shared starting point. It produces the identical resultant as the triangle law — same magnitude, same direction — because the diagonal of the parallelogram and the closing side of the triangle are the same segment. Use the parallelogram picture when both vectors naturally start at one point; use the triangle picture when one motion follows another. ## Adding Vectors With Components In component form, addition is the easy part: **add the matching components**. For \( \vec{P} = P_x\hat{i} + P_y\hat{j} \) and \( \vec{Q} = Q_x\hat{i} + Q_y\hat{j} \), \( \vec{P} + \vec{Q} = (P_x + Q_x)\hat{i} + (P_y + Q_y)\hat{j} \) No angle, no cosine — just line up the \( \hat{i} \) terms and the \( \hat{j} \) terms. This is why component form is the workhorse: the geometric laws explain _why_ the resultant looks the way it does, but the component method is fastest to compute. ## Examples of Vector Addition ### Example 1 **Add \( \vec{P} = 3\hat{i} + 2\hat{j} \) and \( \vec{Q} = \hat{i} + 4\hat{j} \) using components.** \( \vec{P} + \vec{Q} = (3+1)\hat{i} + (2+4)\hat{j} \) **Final answer:** \( 4\hat{i} + 6\hat{j} \). ### Example 2 **Two forces of 6 N and 8 N act on a point at right angles. Find the resultant.** **Correct:** Use the triangle-law magnitude formula with \( \theta = 90° \): ∣R⃗∣=\(\sqrt{6² + 8²} = 10 **Final answer:** the resultant is 10 N.** ### Example 3 **Find the magnitude of the resultant of \( \vec{P} \) (∣P⃗∣=4) and \( \vec{Q} \) (∣Q⃗∣=9) with \( \theta=30° \).** ∣R⃗∣=\(\sqrt{4² + 9² + 2(4)(9)\cos 30°} \) **Final answer:** ∣R⃗∣≈12.62 units. ### Example 4 **Find \( \vec{P} + \vec{Q} \) in 3D for \( \vec{P} = 2\hat{i} - \hat{j} + 3\hat{k} \) and \( \vec{Q} = -\hat{i} + 4\hat{j} - \hat{k} \).** **Final answer:** \( \hat{i} + 3\hat{j} + 2\hat{k} \). ### Example 5 **A hiker walks 3 km east, then 4 km north. How far is the hiker from the start, and in what direction?** **Final answer:** the hiker is 5 km from start, about 53.13° north of east. ### Example 6 **Two equal-magnitude vectors of size m point in the same direction. Find the resultant magnitude.** **Final answer:** 2m. ## Why Vector Addition Matters: "Every Resultant Is A Real Combined Effect" Vector addition isn't a notation trick — it's how the physical world combines influences that have direction. A plane's track over the ground is its airspeed vector plus the wind vector. The net force on a bridge joint is the sum of every cable's tension vector. ## What Are The Most Common Mistakes With Vector Addition? ### Mistake 1: Adding magnitudes directly ### Mistake 2: Joining the vectors the wrong way ### Mistake 3: Sign errors when components are negative ## Conclusion - **Vector addition** combines two vectors into a single resultant that carries their joint effect. - The **triangle law** and the **parallelogram law** give the same answer. - The component method — add matching components — is the fastest in practice. - The resultant magnitude is ∣R⃗∣=P²+Q²+2PQcosθ, which collapses to the Pythagorean result when the vectors are perpendicular. - The biggest mistake is adding magnitudes directly; that only works when the vectors point the same way.