Vector Addition — The Component (Analytical) Method

When you break every vector into x– and y-components, adding vectors feels just like adding ordinary numbers. Here’s the step-by-step story.

1. Writing a vector as components

For a vector in the plane we write
\[ \mathbf A = A_x \,\hat{\mathbf i} + A_y \,\hat{\mathbf j} \] so the numbers \(A_x\) and \(A_y\) tell us “how much” of \(\mathbf A\) points along the x– and y-axes. :contentReference[oaicite:0]{index=0}

2. Adding two vectors in the plane

Give the second vector the same treatment:

\[ \mathbf B = B_x \,\hat{\mathbf i} + B_y \,\hat{\mathbf j}. \]

Add the two vectors term-by-term:

\[ \begin{aligned} \mathbf R &= \mathbf A + \mathbf B \\[2pt] &= (A_x + B_x)\,\hat{\mathbf i} + (A_y + B_y)\,\hat{\mathbf j}. \end{aligned} \] :contentReference[oaicite:1]{index=1}

So the components of the resultant are simply

\[ R_x = A_x + B_x, \qquad R_y = A_y + B_y. \] :contentReference[oaicite:2]{index=2}

3. Stretching to three dimensions

Add a z-component and exactly the same logic works:

\[ \begin{aligned} \mathbf A &= A_x \,\hat{\mathbf i}+A_y \,\hat{\mathbf j}+A_z \,\hat{\mathbf k},\\ \mathbf B &= B_x \,\hat{\mathbf i}+B_y \,\hat{\mathbf j}+B_z \,\hat{\mathbf k},\\ \mathbf R &= (A_x+B_x)\,\hat{\mathbf i}+(A_y+B_y)\,\hat{\mathbf j}+(A_z+B_z)\,\hat{\mathbf k}. \end{aligned} \] :contentReference[oaicite:3]{index=3}

4. Any number of vectors (and even subtraction)

For vectors \(\mathbf a,\mathbf b,\mathbf c\) you can mix “+” and “–” straight across the components:

\[ \mathbf T=\mathbf a+\mathbf b-\mathbf c \; \Longrightarrow \; T_x=a_x+b_x-c_x,\; T_y=a_y+b_y-c_y,\; T_z=a_z+b_z-c_z. \] :contentReference[oaicite:4]{index=4}

5. Magnitude & direction of two-vector combination

When vectors \(\mathbf A\) and \(\mathbf B\) meet at an angle \(\theta\), the length of their sum is given by the Law of Cosines:

\[ R^2 = A^2 + B^2 + 2AB\cos\theta. \] :contentReference[oaicite:5]{index=5}

The direction angles follow the Law of Sines:

\[ \frac{R}{\sin\theta} = \frac{A}{\sin\beta} = \frac{B}{\sin\alpha}, \] where \(\alpha\) and \(\beta\) are the angles between \(\mathbf R\) and \(\mathbf B\), and between \(\mathbf R\) and \(\mathbf A\) respectively. :contentReference[oaicite:6]{index=6}

Sometimes it’s handier to write the direction of \(\mathbf R\) via its tangent:

\[ \tan\alpha=\frac{B\sin\theta}{A+B\cos\theta}. \; :contentReference[oaicite:7]{index=7} \]

6. Quick worked example

Boat + current: A motor-boat heads due north at \(25\text{ km h}^{-1}\). Water flows \(10\text{ km h}^{-1}\) at \(60^\circ\) east of south. Treat the boat’s speed and the current as vectors and use the rules above.
• Magnitude of the result: about \(22\text{ km h}^{-1}\).
• The boat ends up ~\(23^\circ\) off its intended north line. :contentReference[oaicite:8]{index=8}

High-Yield Ideas for NEET

  1. Component trick: \(\displaystyle R_x = A_x + B_x,\; R_y = A_y + B_y\). Easy marks for “find the resultant.” :contentReference[oaicite:9]{index=9}
  2. Law of Cosines for the magnitude of two-vector sums. Expect it in questions that give you two lengths and an included angle. :contentReference[oaicite:10]{index=10}
  3. Law of Sines for the direction of the resultant — handy when the problem asks, “At what angle…?” :contentReference[oaicite:11]{index=11}
  4. Vector addition in 3-D: simply add each axis component; no new formula to learn. :contentReference[oaicite:12]{index=12}
  5. Relative-motion setups (boat–river, aircraft–wind) are nothing more than component addition in disguise. :contentReference[oaicite:13]{index=13}