Constructing a Vector from a Given Point
What is a Vector?
A vector is a mathematical object that has both magnitude (length) and direction. Unlike a simple point or number, a vector tells us how far and which way to go.
📐 Formal Definition
Given a point $A(x_1, y_1)$ and a vector $\vec{v} = \langle a, b \rangle$, we can construct the vector from point $A$ by finding the terminal point $B$ such that:
$$B = (x_1 + a, \; y_1 + b)$$
The vector $\vec{AB}$ then goes from $A$ to $B$, and we write:
$$\vec{AB} = \langle x_2 - x_1, \; y_2 - y_1 \rangle$$
where $A(x_1, y_1)$ is the initial point (tail) and $B(x_2, y_2)$ is the terminal point (head/tip).
🔑 Key Concepts
- Initial Point: The starting point of the vector (the tail).
- Terminal Point: The ending point of the vector (the arrowhead).
- Component Form: $\vec{v} = \langle a, b \rangle$ where $a$ is the horizontal change and $b$ is the vertical change.
- Magnitude: $|\vec{v}| = \sqrt{a^2 + b^2}$
- Direction: $\theta = \tan^{-1}\left(\frac{b}{a}\right)$