Vectors are essential in mathematics for representing quantities with both magnitude and direction. This section explores vector operations and their applications in geometry, enabling a deeper understanding of mathematical concepts and physical phenomena.

Vector Operations

Addition and Subtraction

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Combine vectors by adding or subtracting corresponding components.

• Addition Example: $\mathbf{a} + \mathbf{b} = (3\mathbf{i} + 2\mathbf{j}) + (4\mathbf{i} - \mathbf{j}) = 7\mathbf{i} + \mathbf{j}$
• Subtraction Example: $\mathbf{a} - \mathbf{b} = (3\mathbf{i} + 2\mathbf{j}) - (4\mathbf{i} - \mathbf{j}) = -\mathbf{i} + 3\mathbf{j}$

Multiplication by a Scalar

Alter the magnitude of a vector by multiplying it with a scalar.

• Scalar Multiplication Example: $2\mathbf{a} = 2(3\mathbf{i} + 2\mathbf{j}) = 6\mathbf{i} + 4\mathbf{j}$

Magnitude and Normalisation

• Magnitude Example:
  • For $\mathbf{a} = 3\mathbf{i} + 4\mathbf{j}), (|\mathbf{a}| = \sqrt{3^2 + 4^2} = 5$.
• Unit Vector Example:
  • For $\mathbf{a} = 3\mathbf{i} + 4\mathbf{j}), (\hat{\mathbf{a}} = \frac{1}{5}(3\mathbf{i} + 4\mathbf{j}) = 0.6\mathbf{i} + 0.8\mathbf{j}$.

Geometric Interpretations

Displacement Vector

Represents the shortest route between two points.

Example:

If point A is at (2, 3) and point B is at (5, 7), then $\vec{AB} = (5-2)\mathbf{i} + (7-3)\mathbf{j} = 3\mathbf{i} + 4\mathbf{j}$.

Position Vector

Defines a point's position in relation to the origin.

Example:

If point A is at (2, 3), then $\vec{OA} = 2\mathbf{i} + 3\mathbf{j}$.

Geometrical Principles

Parallelogram Law

Example:

In parallelogram OABC, if $\vec{OA} = 3\mathbf{i} + 2\mathbf{j}$ and $\vec{OC} = \mathbf{i} + 5\mathbf{j}$, then $\vec{OB} = \vec{OA} + \vec{OC} = 4\mathbf{i} + 7\mathbf{j}$.

Midpoint of a Vector Segment

Example:

For a segment $\vec{AB}$ with endpoints A(2, 3) and B(8, 11), the midpoint M is found by $\vec{OM} = \frac{1}{2}[(2\mathbf{i} + 3\mathbf{j}) + (8\mathbf{i} + 11\mathbf{j})] = 5\mathbf{i} + 7\mathbf{j}$.