2.8 Vectors in the Cartesian Coordinate System | Vectors | Physics 11

A vector is a mathematical object used to represent physical quantities that have both a magnitude (size) and a direction. To analyze and perform calculations with vectors, we need a reference framework. The most common framework is the Cartesian coordinate system, which provides a structured way to describe a vector's orientation and length in space.

Scalar And Vector Quantities→

The Cartesian Coordinate System

The Cartesian coordinate system (also known as the rectangular coordinate system) uses perpendicular axes to specify the location of points in space.

Two-Dimensional (2D) System: Two perpendicular axes — the horizontal x-axis and the vertical y-axis — intersect at the origin (0, 0). Any point in this plane is described by an ordered pair (x, y).
Three-Dimensional (3D) System: A third z-axis, perpendicular to both x and y, is added. Any point in space is described by an ordered triple (x, y, z).

Representing Vectors with Components

In the Cartesian system, a vector is broken down (resolved) into components along each axis.

Rectangular Components Of A Vector→

A vector $A$ is written as the sum of its components:

In 2D: $A = A_{x} \hat{i} + A_{y} \hat{j}$
In 3D: $A = A_{x} \hat{i} + A_{y} \hat{j} + A_{z} \hat{k}$

Where:

$A_{x}, A_{y}, A_{z}$ are the scalar magnitudes of the components along the x, y, and z axes.
$\hat{i}, \hat{j}, \hat{k}$ are unit vectors — vectors of magnitude 1 pointing along the positive x, y, and z axes. They indicate direction only.

If a vector $A$ of magnitude $A$ makes an angle $θ$ with the positive x-axis: $A_{x} = A cos θ A_{y} = A sin θ$

Worked Example: A force $F$ of magnitude 10 N acts at 30° above the positive x-axis. $F_{x} = 10 cos 30° = 10 \times \frac{3}{2} \approx 8.66 N$ $F_{y} = 10 sin 30° = 10 \times \frac{1}{2} = 5 N$ So $F = 8.66 \hat{i} + 5 \hat{j}$ N.

Calculating the Magnitude of a Vector

The magnitude of a vector is its length — a scalar quantity calculated using the Pythagorean theorem.

Magnitude in 2D: $∣ A ∣ = A_{x}^{2} + A_{y}^{2}$
Magnitude in 3D: $∣ A ∣ = A_{x}^{2} + A_{y}^{2} + A_{z}^{2}$

Determining the Direction of a Vector

The direction of a vector is described by the angles it makes with the coordinate axes.

Direction in 2D: The angle $θ$ measured counter-clockwise from the positive x-axis: $θ = tan^{- 1} (\frac{A _{y}}{A _{x}})$ (The signs of $A_{x}$ and $A_{y}$ determine the correct quadrant.)

Finding Vector From Its Components→

Direction in 3D — Direction Cosines: The angles the vector makes with each positive axis:
- Angle $α$ with x-axis: $cos α = \frac{A _{x}}{∣ A ∣}$
- Angle $β$ with y-axis: $cos β = \frac{A _{y}}{∣ A ∣}$
- Angle $γ$ with z-axis: $cos γ = \frac{A _{z}}{∣ A ∣}$

Special Vectors

Null Vector ( $0$ ): A vector with zero magnitude and arbitrary direction. Results from operations such as $A - A = 0$ .
Resultant Vector: A single vector that produces the same combined effect as two or more vectors acting together.

Summary Table

Concept	2D Formula	3D Formula
Vector Representation	$A = A_{x} \hat{i} + A_{y} \hat{j}$	$A = A_{x} \hat{i} + A_{y} \hat{j} + A_{z} \hat{k}$
Magnitude	$	\vec
Direction	$θ = tan^{- 1} (\frac{A _{y}}{A _{x}})$	Direction Cosines ( $α, β, γ$ )

The Cartesian Coordinate System

The Cartesian coordinate system (also known as the rectangular coordinate system) uses perpendicular axes to specify the location of points in space.

Two-Dimensional (2D) System: Two perpendicular axes — the horizontal x-axis and the vertical y-axis — intersect at the origin (0, 0). Any point in this plane is described by an ordered pair (x, y).

Three-Dimensional (3D) System: A third z-axis, perpendicular to both x and y, is added. Any point in space is described by an ordered triple (x, y, z).

Representing Vectors with Components

In the Cartesian system, a vector is broken down (resolved) into components along each axis.

A vector

A

is written as the sum of its components:

In 2D:

A = A_{x} \hat{i} + A_{y} \hat{j}

In 3D:

A = A_{x} \hat{i} + A_{y} \hat{j} + A_{z} \hat{k}

Where:

A_{x}, A_{y}, A_{z}

are the scalar magnitudes of the components along the x, y, and z axes.

\hat{i}, \hat{j}, \hat{k}

are unit vectors — vectors of magnitude 1 pointing along the positive x, y, and z axes. They indicate direction only.

If a vector

A

of magnitude

A

makes an angle

θ

with the positive x-axis:

A_{x} = A cos θ A_{y} = A sin θ

Worked Example: A force

F

of magnitude 10 N acts at 30° above the positive x-axis.

F_{x} = 10 cos 30° = 10 \times \frac{3}{2} \approx 8.66 N

F_{y} = 10 sin 30° = 10 \times \frac{1}{2} = 5 N

F = 8.66 \hat{i} + 5 \hat{j}

Determining the Direction of a Vector

The direction of a vector is described by the angles it makes with the coordinate axes.

Direction in 2D: The angle

θ

measured counter-clockwise from the positive x-axis:

θ = tan^{- 1} (\frac{A _{y}}{A _{x}})

(The signs of $A_{x}$ and $A_{y}$ determine the correct quadrant.)

Direction in 3D — Direction Cosines: The angles the vector makes with each positive axis:

Angle $α$ with x-axis: $cos α = \frac{A _{x}}{∣ A ∣}$
Angle $β$ with y-axis: $cos β = \frac{A _{y}}{∣ A ∣}$
Angle $γ$ with z-axis: $cos γ = \frac{A _{z}}{∣ A ∣}$

Concept

2D Formula

3D Formula

Vector Representation

A = A_{x} \hat{i} + A_{y} \hat{j}

A = A_{x} \hat{i} + A_{y} \hat{j} + A_{z} \hat{k}

Magnitude

\vec

Direction

θ = tan^{- 1} (\frac{A _{y}}{A _{x}})

Direction Cosines (

α, β, γ

)