Vectors in the Coordinate Plane: Magnitude and Applications

Vectors in the Coordinate Plane

Magnitude of a Vector

The magnitude (or length) of a vector in the coordinate plane is a fundamental concept in trigonometry and analytic geometry. It is calculated using the Pythagorean Theorem, which relates the components of the vector to its overall length.

• Definition: The magnitude of a vector v with components v = <v1, v2> is the distance from the origin to the point (v1, v2).

• Formula: The magnitude is given by: $|\vec{v}| = \sqrt{v_1^2 + v_2^2}$

• Geometric Interpretation: The vector forms the hypotenuse of a right triangle with legs of lengths |v1| and |v2|.

Example Calculation

• Given: $\vec{v} = \langle 4, 3 \rangle$

• Magnitude: $|\vec{v}| = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5$

Generalization

• For any vector $\vec{v} = \langle a, b \rangle$, the magnitude is: $|\vec{v}| = \sqrt{a^2 + b^2}$

Graphical Representation

Vectors can be represented as arrows in the coordinate plane, starting at the origin and ending at the point (v1, v2). The length of the arrow corresponds to the magnitude of the vector.

• Plotting: Draw a right triangle with legs along the x- and y-axes to visualize the components.

• Magnitude: The length of the hypotenuse is the magnitude of the vector.

Applications

• Magnitude calculations are essential in physics (e.g., force, velocity), engineering, and trigonometry for determining distances and lengths.

Summary Table: Vector Magnitude Formula

Vector	Magnitude Formula
$\langle a, b \rangle$	$\sqrt{a^2 + b^2}$
$\langle v_1, v_2 \rangle$	$\sqrt{v_1^2 + v_2^2}$

Additional info: The magnitude formula is a direct application of the Pythagorean Theorem, which is foundational in both trigonometry and analytic geometry.