Magnitude of a Vector

1. Definition

The magnitude of a vector is the size or length of the vector, independent of its direction.

It is written as:

\(|\vec{A}|\)

and represents how large the quantity is.

2. Geometric Interpretation

A vector can be visualized as an arrow:

• length of the arrow → magnitude
• direction of the arrow → direction

Thus, magnitude corresponds to the distance from the tail to the head of the vector.

3. Magnitude from Components (2D)

If a vector is written as:

\(\vec{A}=(A_x , A_y)\)

its magnitude is found using the Pythagorean relationship:

So,

\(|\vec{A}|=\sqrt{{A_x}^2+{A_y}^2}\)​

4. Magnitude in Three Dimensions

For a vector:

\(\vec{A}=(A_x , A_y , A_z)\)

the magnitude is:

\(|\vec{A}|=\sqrt{{A_x}^2+{A_y}^2+{A_z}^2}\)​

5. Example

Given:

\(\vec{A}\) = (6 , 8)

Magnitude:

\(|\vec{A}=\sqrt{6^2+8^2}=\sqrt{36+64}\)=10

6. Physical Meaning in Mechanics

Magnitude gives the intensity or strength of a vector quantity.

Examples:

• magnitude of velocity → speed
• magnitude of force → strength of force
• magnitude of displacement → straight-line distance from start to end

7. Unit Vector Connection

A unit vector has magnitude equal to 1 and represents direction only.

It is obtained by:

\(\hat{A}=\frac{\vec{A}}{|\vec{A}|}​\)