A vector: a directed segment

Some quantities are described by one number: a temperature, a mass, a time. These are scalar quantities, and the number is called a scalar.

For other quantities you also need their direction:

• a displacement – a hiker walked 5 km; where to? North or east – and she ends up in a different place;
• a force – two people pull a rope tied to a post, each in a different direction;
• a velocity – a plane flies east at 400 km/h, and the wind blows to the north-west.

Such a quantity is a vector quantity, and it is described by a vector.

A vector is a directed segment: a segment whose two ends differ – one is called the tail and the other the head.

• The vector with its tail at $\displaystyle A$ and its head at $\displaystyle B$ is written $\displaystyle \overrightarrow{AB}$. A vector is also written as one letter with a line under it: $\displaystyle \underline{u}$.
• The length of the vector is the length of the segment: $\displaystyle {|\overrightarrow{AB}|}={AB}$.
• Two vectors are equal when they have the same length and the same direction. In the drawing $\displaystyle {\overrightarrow{AB}}={\overrightarrow{CD}}$, and the quadrilateral $\displaystyle ABDC$ is a parallelogram.

So a vector may be moved to another place, as long as it stays parallel to itself and keeps its length – it is the same vector.

• The vector $\displaystyle \overrightarrow{BA}$ is opposite to $\displaystyle \overrightarrow{AB}$: the same length, the reverse direction. We write $\displaystyle {\overrightarrow{BA}}={-\overrightarrow{AB}}$.
• The zero vector $\displaystyle {\overrightarrow{AA}}={\underline{0}}$: its length is 0, and it has no direction.
• Two vectors are parallel (collinear) when they lie on parallel lines or on the same line: in the same direction or in reverse directions.

In the parallelogram $\displaystyle ABCD$ the diagonals meet at the point $\displaystyle O$.

• $\displaystyle {\overrightarrow{AB}}={\overrightarrow{DC}}$: opposite sides are parallel and equal, and in the same direction.
• $\displaystyle {\overrightarrow{AD}}={\overrightarrow{BC}}$.
• $\displaystyle {\overrightarrow{AO}}={\overrightarrow{OC}}$, since the diagonals bisect each other; and $\displaystyle {\overrightarrow{OB}}={-\overrightarrow{OD}}$.
• $\displaystyle \overrightarrow{AB}$ and $\displaystyle \overrightarrow{CD}$ are parallel and of equal length, but they are not equal: $\displaystyle {\overrightarrow{CD}}={-\overrightarrow{AB}}$.

$\displaystyle ABCDEF$ is a regular hexagon with centre $\displaystyle O$.

1. Write all the vectors in the drawing (between two marked points) that are equal to $\displaystyle \overrightarrow{AB}$.
2. Write three vectors opposite to $\displaystyle \overrightarrow{OC}$.
3. Is $\displaystyle {\overrightarrow{AF}}={\overrightarrow{CD}}$? Explain.

In the box $\displaystyle ABCDA'B'C'D'$:

1. Write all the vectors between the box’s vertices that are equal to $\displaystyle \overrightarrow{AA'}$.
2. Write all the vectors between the box’s vertices that are opposite to $\displaystyle \overrightarrow{AB}$.

True or false? Explain, or give a counterexample.

1. If $\displaystyle {|\overrightarrow{AB}|}={|\overrightarrow{CD}|}$, then $\displaystyle {\overrightarrow{AB}}={\overrightarrow{CD}}$.
2. If $\displaystyle {\overrightarrow{AB}}={\overrightarrow{CD}}$, then $\displaystyle {\overrightarrow{AC}}={\overrightarrow{BD}}$.

The edge of a cube $\displaystyle ABCDA'B'C'D'$ is 2 long.
What is the length of the vector $\displaystyle \overrightarrow{AC'}$? Exactly.