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Vector Operations

Vector Types

Vector Magnitude and Direction

Every vector has two main components: magnitude (length) and direction.

Consider the example vector CD\overrightarrow{CD} below.

Vector CDCD
Vector CDCD with a length of 4 cm and a direction of 45 degrees relative to the horizontal line.
  • The magnitude of vector CD\overrightarrow{CD} is 4 cm. This indicates the size or value of the vector. The magnitude of a vector v\vec{v} is usually denoted as v|\vec{v}|. So, CD=4|\overrightarrow{CD}| = 4 cm.
  • The direction of vector CD\overrightarrow{CD} is 45° relative to the horizontal line. This direction is crucial and distinguishes vectors from scalar quantities (which only have magnitude). Direction can be expressed using angles, compass points (like Northeast), or other references.

Negative Vector (Opposite Vector)

A negative vector or opposite vector is a vector that has the same magnitude but opposite direction to the original vector.

Imagine Andi walks 100 m in a direction of 30° (let's call this displacement vector A\vec{A}). Then, Andi returns to the starting position. This second displacement is the opposite vector of A\vec{A}, which we write as A-\vec{A}.

Vector and Opposite Vector
Vector AA and vector A-A have the same magnitude but opposite directions.
  • Vectors A\vec{A} and A-\vec{A} have the same magnitude (A=A|\vec{A}| = |-\vec{A}|).
  • The direction of vector A-\vec{A} is exactly opposite to the direction of vector A\vec{A}. If A\vec{A} points in one direction, A-\vec{A} points in the opposite direction (180° difference).

Zero Vector

The zero vector is a special vector because it has zero magnitude. Due to its zero length, this vector does not have a specific direction.

The zero vector can be visualized as a single point, where the initial point and terminal point coincide. The zero vector is usually denoted by 0\vec{0}.

Example:

If Andi walks 100 m east, then walks back 100 m west, Andi's total displacement is zero. This total displacement can be represented as the zero vector (0\vec{0}).

Equivalent Vectors (Equal Vectors)

Two or more vectors are said to be equivalent or equal if they have the same magnitude (length) and direction, even if their starting points are different.

Consider the graph below showing three equivalent vectors: CD\overrightarrow{CD}, EF\overrightarrow{EF}, and KL\overrightarrow{KL}.

Equivalent Vectors
Vectors CDCD, EFEF, and KLKL have the same magnitude and direction.

The three vectors above have the same magnitude and direction, so they are equivalent. We can write this as:

CD=EF=KL\overrightarrow{CD} = \overrightarrow{EF} = \overrightarrow{KL}

A vector is said to be equivalent to another vector if it has the same magnitude and direction as the other vector.

Exercise

Consider the two vectors below:

Comparison of Vector AA and Vector BB
Comparison of Vector AA and Vector BB

Is vector A\vec{A} the opposite vector of B\vec{B}?

Answer:

Vector A\vec{A} is not the opposite vector of B\vec{B}. To be an opposite vector, two conditions must be met:

  1. They must have the same magnitude: Visually, AB|\vec{A}| \neq |\vec{B}| (their lengths appear different).
  2. Their directions must be exactly opposite (180° apart): The direction of A\vec{A} is not opposite to the direction of B\vec{B}.

Since these two conditions are not met, A\vec{A} is not the opposite vector of B\vec{B}.

How to make vectors A\vec{A} and B\vec{B} opposite?

To make vectors A\vec{A} and B\vec{B} opposite, you could define them like this:

Opposite Vectors AA and BB
Vector AA and BB are opposite