Basic Concepts of Vector Algebra

Vector Algebra begins with the study of quantities that have both magnitude and direction. In everyday life, some quantities need only size for description, while others need both size and direction for complete meaning. This topic builds the foundation for later concepts such as vector operations, geometry in three dimensions, and applications in physics and engineering.

A scalar quantity is a physical quantity that has magnitude only.

A vector quantity is a physical quantity that has magnitude as well as direction.

A vector is a quantity that has magnitude as well as direction. Geometrically, a vector is represented by a directed line segment such as  \[\vec{AB}\], where A is the initial point and B is the terminal point.

The magnitude of vector \[\vec{AB}\] is the length of the directed line segment AB. It is written as \[|\vec{AB}|\], \[|\vec{a}|\], or simply a. The magnitude of a vector is never negative because it represents length.

In three-dimensional geometry, the vector drawn from the origin O(0, 0, 0) to a point P(x, y, z) is called the position vector of the point P. It is written as \[\vec{OP}\]. If point P(x, y, z) is given, then the magnitude of its position vector is:

Magnitude of a Position Vector: Find the magnitude of the position vector of point P(3, 4, 0).

Solution: The position vector is \[\vec{OP}\]. Using the formula,

So, the magnitude of the position vector is 5 units.

• Scalars have only magnitude.

• Vectors have magnitude and direction.

• Vectors are represented by directed line segments.

• \[\vec{AB}\] represents a vector from A to B.

• Magnitude of a vector is its length and is always non-negative.

• \[\vec{OP}\] is the position vector of point \[P(x, y, z)\].

• \[|\vec{OP}| = \sqrt{x^2 + y^2 + z^2}\].