Here's why:

* Pythagorean theorem: When you have two velocity vectors that are perpendicular to each other (like horizontal and vertical components), you can use the Pythagorean theorem to find the magnitude of the resultant velocity. The theorem states: a² + b² = c² where 'c' is the hypotenuse (resultant velocity) and 'a' and 'b' are the two perpendicular components.

* Trigonometry: Trigonometry is used to find the direction of the resultant velocity. You use sine, cosine, or tangent functions to determine the angle of the resultant velocity relative to the horizontal or vertical axis.

Example:

Imagine an object moving with a velocity of 5 m/s to the east and 12 m/s to the north. To find the resultant velocity:

1. Pythagorean Theorem:

* Resultant velocity² = 5² + 12² = 169

* Resultant velocity = √169 = 13 m/s

2. Trigonometry:

* To find the angle, we can use the tangent function: tan(θ) = opposite/adjacent = 12/5

* θ = arctan(12/5) ≈ 67.38°

Therefore, the resultant velocity is 13 m/s at an angle of approximately 67.38° north of east.