What is a vector?

A vector is a quantity that has a magnitudeLength or strength of a vector. and a direction. Vectors play an important role in various fields, including physics, engineering, computer graphics, and machine learning, where they are used for calculations and modeling of physical phenomena and mathematical operations.

Representation of a vector

Let's suppose we have a vector $v$. We can represent this vector as $\vec{v}$. To represent it graphically, we draw an arrow consisting of the following:

• Head: The beginning of the arrow.

• Tail: The end of the arrow.

• Magnitude: The length of the vector.

• Direction: The direction of the vector.

Vectors in two-dimensional space

A two-dimensional vector is a vector that has direction in two-dimensional space: x-axis and y-axis. Suppose we have a vector $\vec{v}$, we can represent it in two dimensional space as:

• $x$ the magnitude along the x-axis

• $y$ is the magnitude along the y-axis

Example

Let's consider a vector $\vec{v}$ with components (4,3). This means that $\vec{v}$ has a magnitude of 4 units along the x-axis and 3 units along the y-axis. Now we can represent it in a diagram:

Vectors in three-dimensional space

A three-dimensional vector is a vector that has direction in three-dimensional space: x-axis, y-axis, and z-axis. Suppose we have a vector $\vec{v}$ we can represent it in three-dimensional space as:

• $x$ the magnitude along the x-axis

• $y$ is the magnitude along the y-axis

• $z$ is the magnitude along the z-axis

Example

Let's consider a vector $\vec{v}$ with components (5, 4, 2). This means that $\vec{v}$ has a magnitude of 5 units along the x-axis, 4 units along the y-axis, and 2 units along the z-axis. Now, let's represent it in a diagram.

Real-life examples

Let's discuss some real-life examples of vectors now.

• Force: A vector showing the strength and direction of applied force on an object.

• Velocity: A vector representing the speed and direction of a moving object.

• Weight: A vector showing the magnitude and direction of gravity's force on an object.

Conclusion

Vectors are mathematical quantities that represent both magnitude and direction. They help describe physical quantities such as displacement, velocity, and force in various fields of science and engineering. Moreover, they provide a concise and powerful way to represent and analyze complex geometric and physical concepts.