# What is a Vector? Definition and Examples

What is a vector? A vector, also called vector quantity, is a quantity that has both magnitude and direction. For example, suppose you are pulling a cart up a 25 degrees incline and it requires an effort of 75 pounds.

This situation is describing a vector because it has a magnitude and it has a direction. The magnitude is 75 pounds and the direction is 30 degrees from the horizontal.

## What is a vector geometrically?

Geometrically, a vector is a directed line segment with an initial point and a terminal point such as the one shown below.

We can use boldface letters such as **v**, **w**, **s** to denote vectors. Notice the boldface letter **v** used above to denote the vector shown in blue.

## Relationship between two vectors

There are four possible relationships between two vectors **v** and **w**.

1.

**v** = **w **because the vectors have the same magnitude and the same direction**.**

2.

**v** ≠ **w **because the vectors have the same direction, but different magnitudes.

3.

**v** ≠ **w **because the vectors have the same magnitudes, but different directions.

4.

**v** ≠ **w **because the vectors have the same magnitudes, but opposite directions.

## More real life examples of vectors

- Driving a car due south at 65 miles per hour.

- Picking a box from the floor and lifting it straight up or at an angle.

- Kicking a football with a speed of 38 feet per second at an angle of 50 degrees.

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