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= Vector (Physics): Definition and Mathematical Representation =

== Introduction ==
In physics, a '''vector''' is a quantity that has both '''magnitude''' and '''direction'''. Vectors are essential in describing physical phenomena such as displacement, velocity, acceleration, force, and momentum.

Unlike scalars, which are described by a single value, vectors are represented by arrows whose length corresponds to magnitude and whose orientation indicates direction.

== Definition ==

A vector <math>\vec{A}</math> in component form is written as:

<math>
\vec{A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k}
</math>

Where:
* <math>A_x, A_y, A_z</math> are the components of the vector along the x, y, and z axes respectively.
* <math>\hat{i}, \hat{j}, \hat{k}</math> are the unit vectors in the x, y, and z directions.

== Magnitude of a Vector ==

The magnitude (length) of a 3D vector is given by:

<math>
|\vec{A}| = \sqrt{A_x^2 + A_y^2 + A_z^2}
</math>

For a 2D vector:

<math>
|\vec{A}| = \sqrt{A_x^2 + A_y^2}
</math>

== Direction of a Vector ==

The direction (angle <math>\theta</math>) in 2D from the x-axis is:

<math>
\theta = \tan^{-1} \left( \frac{A_y}{A_x} \right)
</math>

== Vector Operations ==

=== 1. Addition ===

<math>
\vec{C} = \vec{A} + \vec{B} = (A_x + B_x)\hat{i} + (A_y + B_y)\hat{j}
</math>

Graphically represented using the **head-to-tail** method or **parallelogram rule**.

=== 2. Subtraction ===

<math>
\vec{C} = \vec{A} - \vec{B} = (A_x - B_x)\hat{i} + (A_y - B_y)\hat{j}
</math>

Equivalent to adding the negative of a vector.

=== 3. Scalar Multiplication ===

<math>
k\vec{A} = (k A_x)\hat{i} + (k A_y)\hat{j}
</math>

Where <math>k</math> is a scalar. If <math>k < 0</math>, the vector direction is reversed.

=== 4. Dot Product (Scalar Product) ===

<math>
\vec{A} \cdot \vec{B} = |\vec{A}||\vec{B}|\cos\theta = A_x B_x + A_y B_y + A_z B_z
</math>

Result is a scalar.

=== 5. Cross Product (Vector Product) ===

<math>
\vec{A} \times \vec{B} = |\vec{A}||\vec{B}|\sin\theta\, \hat{n}
</math>

Where <math>\hat{n}</math> is a unit vector perpendicular to the plane of <math>\vec{A}</math> and <math>\vec{B}</math>. Result is a vector.

== Unit Vectors ==

Unit vectors have magnitude 1 and indicate direction only. Common unit vectors are:
* <math>\hat{i}</math> along x-axis
* <math>\hat{j}</math> along y-axis
* <math>\hat{k}</math> along z-axis

Example: If <math>\vec{v} = 3\hat{i} + 4\hat{j}</math>, then:

<math>
|\vec{v}| = \sqrt{3^2 + 4^2} = 5
</math>

== Applications in Physics ==

Vectors are used to represent:
* [[Displacement]]
* [[Velocity]]
* [[Acceleration]]
* [[Force]]
* [[Momentum]]
* [[Electric Field]] and [[Magnetic Field]]

== Graphical Representation ==

Vectors are shown as arrows:
* Length = magnitude
* Angle = direction
* Arrows can be added/subtracted graphically

== See Also ==
* [[Scalar (physics)]]
* [[Displacement]]
* [[Force]]
* [[Vector Addition]]
* [[Dot Product]]
* [[Cross Product]]
* [[Unit Vector]]
* [[Kinematics]]