Vector addition — triangle law, parallelogram law, component method comparison

Question

Compare the three methods of vector addition: triangle law, parallelogram law, and component method. When should you use each one? Find the resultant of two forces of 3 N and 4 N at 90° to each other.

(CBSE 11 boards ask the laws; JEE Main tests the component method in projectile/force problems)

Solution — Step by Step

Place the tail of the second vector at the head of the first. The resultant is the vector from the tail of the first to the head of the second.

This gives the magnitude using the cosine rule:

R = \sqrt{A^2 + B^2 + 2AB\cos\theta}

where $\theta$ is the angle between the two vectors.

Place both vectors with their tails at the same point. Complete the parallelogram. The diagonal from the common point is the resultant.

Same formula applies: $R = \sqrt{A^2 + B^2 + 2AB\cos\theta}$

Direction: $\tan\alpha = \frac{B\sin\theta}{A + B\cos\theta}$ where $\alpha$ is the angle of resultant with $\vec{A}$ .

Resolve each vector into x and y components:

R_x = A_x + B_x, \quad R_y = A_y + B_y

R = \sqrt{R_x^2 + R_y^2}, \quad \tan\theta = \frac{R_y}{R_x}

This method works for any number of vectors and is the standard approach for JEE problems.

Two forces: 3 N and 4 N at 90°.

R = \sqrt{3^2 + 4^2 + 2(3)(4)\cos 90°} = \sqrt{9 + 16 + 0} = \sqrt{25} = \mathbf{5 \text{ N}}

Direction: $\tan\alpha = \frac{4\sin 90°}{3 + 4\cos 90°} = \frac{4}{3}$ , so $\alpha = 53.1°$ from the 3 N force.

flowchart TD
    A["Need to add vectors"] --> B{How many vectors?}
    B -->|Two| C{Angle given directly?}
    C -->|Yes| D["Triangle/Parallelogram law<br/>R = √(A²+B²+2ABcosθ)"]
    C -->|"Components given"| E["Component method<br/>Rx = Ax+Bx, Ry = Ay+By"]
    B -->|"Three or more"| F["Component method<br/>(most efficient)"]

Why This Works

Vectors are mathematical objects that have both magnitude and direction. You cannot simply add their magnitudes (unless they are in the same direction). The parallelogram and triangle laws account for direction by using geometry. The component method converts the geometric problem into simple arithmetic by working along perpendicular axes.

The component method is preferred in competitive exams because it scales to any number of vectors and avoids the geometric complexity of drawing accurate diagrams.

Alternative Method

Special cases worth memorising: when two equal vectors of magnitude $A$ are at angle $\theta$ , the resultant is $R = 2A\cos(\theta/2)$ . At $\theta = 60°$ : $R = A\sqrt{3}$ . At $\theta = 90°$ : $R = A\sqrt{2}$ . At $\theta = 120°$ : $R = A$ . At $\theta = 180°$ : $R = 0$ . These shortcuts save time in MCQs.

Common Mistake

Students add vector magnitudes directly: “3 N + 4 N = 7 N.” This is only correct when vectors are in the same direction ( $\theta = 0°$ ). At 90°, the resultant is 5 N (not 7 N). At 180° (opposite directions), it would be 1 N (not 7 N). The resultant of two vectors always lies between $|A-B|$ and $A+B$ . If your answer is outside this range, you have made an error.