Vector Addition

Vector addition is the process of finding a single vector, known as the resultant, which represents the combined effect of two or more individual vectors. If vector $\mathbf{a}$ represents one movement and vector $\mathbf{b}$ represents a subsequent movement, the sum $\mathbf{a} + \mathbf{b}$ represents the direct path from the very start to the very end.

The resultant vector is the diagonal of a parallelogram or the third side of a triangle formed by the original vectors. It effectively replaces multiple steps with one single equivalent displacement.

The zero vector, denoted as $\mathbf{0}$, is the result of adding a vector to its exact opposite (e.g., $\vec{PQ} + \vec{QP} = \mathbf{0}$). It represents a state of no net movement or a return to the original starting position.

Algebraic Addition (Column Vectors)

• To add vectors in column form, simply sum the corresponding top (x) and bottom (y) components independently.
• For example, if $\mathbf{a} = \begin{pmatrix} x_1 \\ y_1 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} x_2 \\ y_2 \end{pmatrix}$, then $\mathbf{a} + \mathbf{b} = \begin{pmatrix} x_1 + x_2 \\ y_1 + y_2 \end{pmatrix}$.

Algebraic Addition (Unit Vector Notation)

• When using $\mathbf{i}$ and $\mathbf{j}$ notation, group the like terms together and sum their coefficients.
• The sum of $(x_1\mathbf{i} + y_1\mathbf{j})$ and $(x_2\mathbf{i} + y_2\mathbf{j})$ is calculated as $(x_1 + x_2)\mathbf{i} + (y_1 + y_2)\mathbf{j}$.

Geometric Addition (Nose-to-Tail)

• Place the 'tail' of the second vector at the 'nose' (arrowhead) of the first vector.
• The resultant vector is drawn from the tail of the first vector to the nose of the last vector in the sequence.

It is vital to distinguish between vector addition and scalar multiplication, as they affect the vector's properties differently.

• Position Vectors vs. Displacement Vectors: A position vector $\vec{OA}$ always starts at the origin $(0,0)$, while a displacement vector $\vec{AB}$ represents the movement between any two points. The relationship is defined by $\vec{AB} = \vec{OB} - \vec{OA}$.

• Always draw a sketch: Even a rough diagram can prevent 'direction errors' where you accidentally subtract a vector instead of adding it. Visualizing the 'journey' helps verify if the resultant's direction makes physical sense.

• Check your signs: When adding column vectors with negative components, ensure you apply standard arithmetic rules (e.g., $5 + (-3) = 2$). This is the most common source of lost marks in vector algebra.

• Verify Parallelism: If an exam question asks you to prove two lines are parallel, calculate the vectors for both lines and check if one is a scalar multiple of the other (e.g., $\mathbf{a} = k\mathbf{b}$).

• Resultant Magnitude: Remember that the magnitude of the resultant $|\mathbf{a} + \mathbf{b}|$ is NOT generally equal to $|\mathbf{a}| + |\mathbf{b}|$. You must find the resultant vector first, then use Pythagoras' theorem to find its magnitude.

• The 'Scalar Sum' Error: Students often mistakenly assume they can add the magnitudes of two vectors to find the magnitude of the sum. This only works if the vectors are pointing in the exact same direction; otherwise, the triangle inequality applies.

• Incorrect Subtraction Order: When finding the vector between two points $A$ and $B$, students often calculate $\mathbf{a} - \mathbf{b}$ instead of the correct $\mathbf{b} - \mathbf{a}$. Always remember: $\vec{AB} = \text{Destination} - \text{Start}$.

• Mixing Components: Ensure you never add an $\mathbf{i}$ component to a $\mathbf{j}$ component. They are perpendicular unit vectors and must remain separate in the final expression.