Magnitude

Definition

Magnitude represents the size or length of a vector. It is always a non-negative value and can be found using the Pythagorean theorem in two or three dimensions.

5 Must Know Facts For Your Next Test

1. The magnitude of a vector $\mathbf{v} = (v_1, v_2)$ in two dimensions is given by $|\mathbf{v}| = \sqrt{v_1^2 + v_2^2}$.
2. For a vector $\mathbf{u} = (u_1, u_2, u_3)$ in three dimensions, its magnitude is calculated as $|\mathbf{u}| = \sqrt{u_1^2 + u_2^2 + u_3^2}$.
3. In polar coordinates, the magnitude of a complex number $z = r(\cos(\theta) + i \sin(\theta))$ is simply $r$.
4. Magnitude is always non-negative and equals zero if and only if the vector itself is the zero vector.
5. When transforming functions graphically, magnitudes can represent scaling factors for stretching or compressing graphs.

Review Questions

• How do you calculate the magnitude of a vector in two-dimensional space?
• What is the relationship between the magnitude of a complex number in polar form and its representation?
• Why is the magnitude of any vector always non-negative?

Related terms

Vector: A quantity having both direction and magnitude, often represented as an arrow from one point to another.

Polar Coordinates: A coordinate system where each point on a plane is determined by an angle and a distance from a reference point.

Complex Numbers: Numbers that have both real and imaginary parts, usually written in the form $a + bi$.