1. Place the Tail of the Second Vector at the Head of the First Vector: Arrange the vectors so that the head of one vector touches the tail of the next.
2. Draw the Resultant Vector: The resultant vector (sum) is drawn from the tail of the first vector to the head of the last vector.
3. Measure the Magnitude and Direction: The length of the resultant vector represents its magnitude, and the direction is the angle it makes with a reference axis.

Example: Consider two vectors $\mathbf{A}$ and $\mathbf{B}$:

• $\mathbf{A}$ is 3 units long, pointing east.
• $\mathbf{B}$ is 4 units long, pointing north. To add them:

4. Draw vector $\mathbf{A}$ horizontally (eastward).
5. Draw vector $\mathbf{B}$ starting from the head of $\mathbf{A}$ and pointing vertically (northward).
6. The resultant vector $\mathbf{R}$ is the diagonal of the right triangle formed, starting from the tail of $\mathbf{A}$ to the head of $\mathbf{B}$. The magnitude of $\mathbf{R}$ can be found using the Pythagorean theorem:

$|\mathbf{R}| = \sqrt{|\mathbf{A}|^2 + |\mathbf{B}|^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = :highlight[5 units]$

This method involves breaking each vector into its components along the x and y axes, then adding the corresponding components to find the components of the resultant vector. 7. Resolve Each Vector into Components:

• For vector $\mathbf{A} = (A_x, A_y)$ and vector $\mathbf{B} = (B_x, B_y)$, where $A_x$ and $A_y$ are the components of $\mathbf{A}$ along the $x$ and $y$ axes, respectively.

8. Add the Corresponding Components:

• The $x$-component of the resultant vector $\mathbf{R}$ is $R_x = A_x + B_x$.
• The $y$-component of the resultant vector $\mathbf{R}$ is $R_y = A_y + B_y$.

9. Form the Resultant Vector:

• The resultant vector is $\mathbf{R} = (R_x, R_y)$.

10. Calculate the Magnitude and Direction:

• Magnitude: $|\mathbf{R}| = \sqrt{R_x^2 + R_y^2}$
• Direction (angle $\theta$ with the $x$-axis): $\theta = \tan^{-1}\left(\frac{R_y}{R_x}\right)$

Example: Add the following vectors:

• $\mathbf{A} = (3, 4)$
• $\mathbf{B} = (1, -2)$ Solution:

11. Resolve into Components:

• $\mathbf{A}$ has components $A_x = 3$ and $A_y = 4$ .
• $\mathbf{B}$ has components $B_x = 1$ and $B_y = -2$ .

12. Add the Components:

• $R_x = A_x + B_x = 3 + 1 = :highlight[4]$
• $R_y = A_y + B_y = 4 - 2 = :highlight[2]$

13. Form the Resultant Vector:

• The resultant vector is $\mathbf{R} = (4, 2)$.

14. Calculate Magnitude and Direction:

• Magnitude: $|\mathbf{R}| = \sqrt{4^2 + 2^2} = \sqrt{16 + 4} = \sqrt{20} = :highlight[2\sqrt{5} ≈ 4.47]$
• Direction: $\theta = \tan^{-1}\left(\frac{2}{4}\right) = \tan^{-1}(0.5) ≈ :highlight[26.57°]$ (above the positive $x$-axis).

• Vector addition can be done geometrically by placing vectors head to tail and drawing the resultant, or algebraically by adding their components.
• The geometric method provides a visual understanding, while the algebraic method is precise and useful for complex problems.
• The resultant vector's magnitude and direction provide the combined effect of the original vectors.