Define vectors, discuss vector representation, operations (addition, subtraction, scalar multiplication), magnitude, unit vectors, and direction.
Introduction to Vectors
Key concepts, definitions, and operations for a solid foundation in vector mathematics.
| Aspect | Description |
|---|---|
| Definition | A quantity that has both magnitude (size) and direction. |
| Notation | Usually boldface v, an arrow (\vec{v}), or a letter with a hat for a unit vector (\hat{v}). |
| Examples | Displacement, velocity, acceleration, force, electric field. |
| Contrast with Scalars | Scalars have only magnitude (e.g., mass, temperature, time). |
For a vector (\vec{v}= \langle v_x, v_y, v_z\rangle):
[ |\vec{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2} ]
| Operation | Symbol | Formula (3‑D) | Geometric Meaning |
|---|---|---|---|
| Addition | (\vec{u}+\vec{v}) | (\langle u_x+v_x,; u_y+v_y,; u_z+v_z\rangle) | “Tip‑to‑tail” (parallelogram) |
| Subtraction | (\vec{u}-\vec{v}) | (\langle u_x-v_x,; u_y-v_y,; u_z-v_z\rangle) | Vector from the head of v to the head of u |
| Scalar Multiplication | (c\vec{v}) | (\langle c,v_x,; c,v_y,; c,v_z\rangle) | Stretches/compresses magnitude; reverses direction if (c<0) |
| Dot (Scalar) Product | (\vec{u}\cdot\vec{v}) | (u_xv_x+u_yv_y+u_zv_z = | \ |