Concept of a Vector

Introduction

A vector is a mathematical entity that has both magnitude and direction. Vectors are fundamental in mathematics and physics for solving problems related to position, displacement, forces, and other directional quantities.

Vectors can be represented geometrically as directed line segments and algebraically using components in a coordinate system.

Definition of a Vector

A vector is represented as a directed line segment, where:

• The length of the segment represents the magnitude of the vector.
• The arrowhead represents the direction of the vector.

Position Vectors and Displacement Vectors

Position Vector: The position vector describes the position of a point relative to an origin.

For example, the position vectors of points \( A \) and \( B \) are:

\( \vec{OA} = \vec{a} \) and \( \vec{OB} = \vec{b} \)

Displacement Vector: The displacement vector \( \vec{AB} \) from point \( A \) to point \( B \) is calculated as:

\(\vec{AB} = \vec{b} – \vec{a}\)

Representation of Vectors Using Directed Line Segments

Vectors are often represented graphically as arrows, where:

• The tail of the arrow is the starting point.
• The head of the arrow is the terminal point.

The length of the arrow corresponds to the magnitude, and the direction is indicated by the arrowhead.

Components of a Vector

A vector in three-dimensional space can be written in component form:

\[ \vec{v} = \begin{bmatrix} v_1 \\ v_2 \\ v_3 \end{bmatrix} = v_1 \vec{i} + v_2 \vec{j} + v_3 \vec{k} \]

• \( v_1 \), \( v_2 \), and \( v_3 \) are the components of the vector along the x, y, and z axes.
• \(\vec{i}\), \(\vec{j}\), and \(\vec{k}\) are the unit base vectors along the x, y, and z axes, respectively.

Algebraic and Geometric Operations on Vectors

Sum and Difference of Two Vectors

The sum of two vectors \( \vec{u} \) and \( \vec{v} \) is obtained by adding their components:

\[ \vec{u} + \vec{v} = \begin{bmatrix} u_1 + v_1 \\ u_2 + v_2 \\ u_3 + v_3 \end{bmatrix} \]

Graphically, the sum is represented using the triangle law or the parallelogram law of vector addition.

The difference of two vectors \( \vec{u} \) and \( \vec{v} \) is given by:

\[ \vec{u} – \vec{v} = \begin{bmatrix} u_1 – v_1 \\ u_2 – v_2 \\ u_3 – v_3 \end{bmatrix} \]

Zero Vector and Negative Vector

• Zero Vector (\(\vec{0}\)): A vector with zero magnitude and no specific direction. \(\vec{0} = [0, 0, 0]\).
• Negative Vector (\(-\vec{v}\)): A vector of the same magnitude as \( \vec{v} \) but in the opposite direction.

Multiplication of a Vector by a Scalar

When a vector \( \vec{v} \) is multiplied by a scalar \( k \):

\[ k \vec{v} = \begin{bmatrix} k v_1 \\ k v_2 \\ k v_3 \end{bmatrix} \]

• If \( k > 0 \), the direction remains the same.
• If \( k < 0 \), the direction is reversed.

Vectors that are scalar multiples of each other are called parallel vectors.

Magnitude of a Vector

The magnitude (or length) of a vector \( \vec{v} = [v_1, v_2, v_3] \) is given by:

\[ |\vec{v}| = \sqrt{v_1^2 + v_2^2 + v_3^2} \]

Unit Vectors

A unit vector has a magnitude of 1 and indicates the direction of a vector. It is obtained by dividing the vector by its magnitude:

\[ \vec{u} = \frac{\vec{v}}{|\vec{v}|} \]

Distance Between Two Points

The distance between two points \( A(x_1, y_1, z_1) \) and \( B(x_2, y_2, z_2) \) is the magnitude of the displacement vector \( \vec{AB} \):

\[ |\vec{AB}| = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2 + (z_2 – z_1)^2} \]