Vectors

Vectors

Basic Concepts

• A vector is a quantity that has both magnitude (size) and direction.
• Vectors are represented by an arrow. The length of the arrow shows the magnitude and the direction of the arrow shows the direction of the vector.
• Vectors can exist in any number of dimensions. However, for the purpose of this level, mainly 2D and 3D vectors are considered.

Vector Addition and Subtraction

• Vector addition is done by a process called ‘tip-to-tail’ addition. The second vector starts where the first vector ends.
• To subtract a vector, simply reverse the direction of the vector to be subtracted and then add.

Scalar Multiplication

• Vectors can be multiplied by scalars (just ordinary numbers). Scalar multiplication changes the magnitude of the vector, but the direction stays the same unless multiplied by a negative scalar.

Vector Components

• Generally, a vector is broken down into its components, often based on standard coordinate directions, e.g., x and y components in 2D, x, y and z in 3D.

Dot Product

• The dot product of two vectors is a scalar quantity that gives the magnitude of one vector times the magnitude of the second vector times the cosine of the angle between them.

Cross Product

• The cross product of two vectors is a vector whose magnitude is the area of the parallelogram that the vectors form, and its direction is perpendicular to the plane of the parallelogram.

Practical Applications

• Vectors are commonly used in physics to describe forces, velocity and acceleration.
• In computer graphics, vectors are used to calculate how 3D models are rotated and scaled on the screen.

Remember that vectors can be used to describe any quantity that has both a magnitude and direction. This opens up a wide range of practical applications in the real world.