Introduction to Geometric Vectors

Geometric Vectors

Understanding Vectors in the Coordinate Plane

Vectors are mathematical objects that have both magnitude (length) and direction. In the coordinate plane, vectors are often represented as directed line segments from one point to another.

• Definition: A vector v from point A to point B is written as v = (v1, v2), where v1 and v2 are the horizontal and vertical components, respectively.

• Notation: If A = (x1, y1) and B = (x2, y2), then the vector from A to B is: $\vec{AB} = (x_2 - x_1,\ y_2 - y_1)$

• Magnitude of a Vector: The length of vector v = (a, b) is: $|\vec{v}| = \sqrt{a^2 + b^2}$

Example Problem

Given: Points A (2, 4) and B (7, 1). Find the vector from A to B.

• Step 1: Subtract the coordinates: $\vec{AB} = (7 - 2,\ 1 - 4) = (5, -3)$

• Step 2: Write the vector in component form: $\vec{AB} = 5\hat{i} - 3\hat{j}$

• Step 3: Find the magnitude: $|\vec{AB}| = \sqrt{5^2 + (-3)^2} = \sqrt{25 + 9} = \sqrt{34}$

Applications

• Vectors are used to represent physical quantities such as force, velocity, and displacement.

• They are fundamental in physics, engineering, and computer graphics for describing motion and direction.