Unit Vectors and Magnitude in Trigonometry

Unit Vectors and Magnitude

Introduction to Unit Vectors

Unit vectors are vectors with a magnitude of 1, used to indicate direction in space. In trigonometry and physics, any vector can be expressed as a combination of unit vectors along the coordinate axes. The process of finding a unit vector in the direction of a given vector involves dividing the vector by its magnitude.

• Unit Vector: A vector of length 1 in a specified direction.

• Magnitude of a Vector: The length or size of the vector, calculated using the Pythagorean theorem in two or three dimensions.

Finding the Unit Vector

Given a vector v = <a, b>, the unit vector u in the direction of v is found by:

• Calculate the magnitude: $|\vec{v}| = \sqrt{a^2 + b^2}$

• Divide each component by the magnitude: $\vec{u} = \left\langle \frac{a}{|\vec{v}|}, \frac{b}{|\vec{v}|} \right\rangle$

Examples

• Example 1: Find a unit vector in the direction of $\vec{v} = \langle 6, 8 \rangle$.

  • Magnitude: $|\vec{v}| = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$

  • Unit vector: $\vec{u} = \left\langle \frac{6}{10}, \frac{8}{10} \right\rangle = \langle 0.6, 0.8 \rangle$

• Example 2: Find a unit vector in the direction of $\vec{v} = \langle -3, 12 \rangle$.

  • Magnitude: $|\vec{v}| = \sqrt{(-3)^2 + 12^2} = \sqrt{9 + 144} = \sqrt{153}$

  • Unit vector: $\vec{u} = \left\langle \frac{-3}{\sqrt{153}}, \frac{12}{\sqrt{153}} \right\rangle$

Practice Problems

1. Find a unit vector in the direction of $\vec{v} = \langle 4, 3 \rangle$.

2. Find a unit vector in the direction of $\vec{v} = \langle 5, -12 \rangle$.

3. Find a unit vector in the direction of $\vec{v} = \langle 2, -2 \rangle$.

Applications: Unit vectors are essential in physics for representing directions of forces, velocities, and other vector quantities. In trigonometry, they are used to express directions in coordinate geometry and to simplify vector calculations.