1. Meaning of Scalars and Vectors
A scalar is a quantity that has only magnitude. Examples include length, mass, temperature, and time.
A vector is a quantity that has both magnitude and direction. Examples include displacement, velocity, force, and acceleration.
The key difference is that scalars tell you “how much,” while vectors tell you “how much” and “in which direction.”
2. Representation of a Vector
A vector is usually shown as an arrow. The arrow has two parts:
- Tail – where the vector starts
- Head – where the vector ends
The length of the arrow shows its magnitude, and the direction of the arrow shows its direction.
If a vector goes from point \(A\) to point \(B\), we write it as \(\vec{AB}\).
Vectors are often written in bold (\(\mathbf{v}\)) or with a small arrow on top (\(\vec{v}\)).
2.1. Notation for Vectors
Common notations include:
- \(\vec{v}\)
- \(\vec{AB}\)
- \(\mathbf{v}\)
In coordinate form, a vector in two dimensions is written as \(\langle x, y \rangle\), and in three dimensions as \(\langle x, y, z \rangle\).
3. Vector Equality
Two vectors are considered equal if they have:
- The same magnitude
- The same direction
Their position does not matter. Even if one vector starts at a different place, they are equal as long as their length and direction match.
3.1. Example of Equal Vectors
If one arrow goes 3 units to the right and another arrow somewhere else also goes 3 units to the right, the two represent equal vectors.
Symbolically:
\[ \vec{u} = \vec{v} \quad \text{if they have the same magnitude and direction.} \]
4. Examples of Vectors in Everyday Geometry
Some simple vector ideas around us:
- A person walking 5 meters north represents a vector with magnitude 5 and direction north.
- A force of 10 newtons pushing an object along a slope is a vector with magnitude 10 and the direction of the slope.
- Movement from point \(A(x_1, y_1)\) to point \(B(x_2, y_2)\) forms the vector:
\[ \vec{AB} = \langle x_2 - x_1,\; y_2 - y_1 \rangle \]