Unit Vectors and Direction in Trigonometry

Unit Vectors and Direction

Definition of a Unit Vector

A unit vector is a vector with a magnitude (length) of 1. Unit vectors are used to indicate direction only, without regard to magnitude.

• Notation: A unit vector in the direction of vector v is often written as û or vunit.

• Formula: To find the unit vector in the direction of a given vector v, divide the vector by its magnitude:

$\mathbf{u} = \frac{\mathbf{v}}{\|\mathbf{v}\|}$

Finding the Magnitude of a Vector

The magnitude (or length) of a vector v = <a, b> is calculated using the Pythagorean theorem:

$\|\mathbf{v}\| = \sqrt{a^2 + b^2}$

Example:

• Given v = <4, 3>, find the magnitude:

$\|\mathbf{v}\| = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5$

Finding the Unit Vector in the Direction of v

• Given v = <4, 3>, and its magnitude is 5, the unit vector is:

$\mathbf{u} = \frac{1}{5}\langle 4, 3 \rangle = \langle \frac{4}{5}, \frac{3}{5} \rangle$

Graphical Representation:

• The vector v is shown as an arrow from the origin to the point (4, 3) on the coordinate plane.

• The unit vector points in the same direction but has a length of 1.

Expressing a Vector as a Scalar Multiple of a Unit Vector

Any nonzero vector v can be written as the product of its magnitude and its unit vector:

$\mathbf{v} = \|\mathbf{v}\| \cdot \mathbf{u}$

Example:

• Given v = <4, 3>, and its unit vector u = <4/5, 3/5>, then:

$\mathbf{v} = 5 \cdot \langle \frac{4}{5}, \frac{3}{5} \rangle = \langle 4, 3 \rangle$

Solving for a Scalar Multiple

• To express a vector v as a scalar multiple of a unit vector c:

$\mathbf{v} = k \mathbf{c}$

• Given v = <4, 3> and c = <2/\sqrt{13}, 3/\sqrt{13}>, solve for k:

$\langle 4, 3 \rangle = k \langle \frac{2}{\sqrt{13}}, \frac{3}{\sqrt{13}} \rangle$

Set up equations for each component:

• $4 = k \cdot \frac{2}{\sqrt{13}}$

• $3 = k \cdot \frac{3}{\sqrt{13}}$

Solve for k:

• $k = 4 \cdot \frac{\sqrt{13}}{2} = 2\sqrt{13}$

• $k = 3 \cdot \frac{\sqrt{13}}{3} = \sqrt{13}$

Check for consistency; if both components yield the same k, the solution is valid.

Additional info: This process is fundamental in trigonometry and vector analysis, especially in physics and engineering applications where direction and magnitude must be separated.