91Ó°ÊÓ

Vectors have both magnitude and direction. The magnitude of a vector, sometimes referred to as its 'length', quantifies how long or large the vector is.
For a vector \( \mathbf{u} = \langle a, b \rangle \), its magnitude is calculated using the formula:

• \( \| \mathbf{u} \| = \sqrt{a^2 + b^2} \)

This formula comes from the Pythagorean theorem, where the vector acts like the hypotenuse of a right triangle; \( a \) and \( b \) are the perpendicular sides.
In the case of the vector \( \mathbf{u} = \langle -5, -5 \rangle \), substituting these values into the formula gives us \( \| \mathbf{u} \| = \sqrt{(-5)^2 + (-5)^2} = 5\sqrt{2} \).
Therefore, the vector's magnitude is \( 5\sqrt{2} \). This tells us how far the vector extends in space from its initial point.

The direction angle of a vector is crucial as it describes the vector's orientation. Calculating the direction angle involves the tangent function.
The direction angle \( \theta \) of a vector \( \mathbf{u} = \langle a, b \rangle \) is often found using:

• \( \theta = \tan^{-1} \left( \frac{b}{a} \right) \)

This angle is determined from the positive x-axis in a counterclockwise direction. In mathematics, direction angles provide a way to represent vector orientation along with magnitude.
For the vector \( \mathbf{u} = \langle -5, -5 \rangle \), we calculate \( \theta = \tan^{-1}(1) = 45^\circ \). However, because vectors in the third quadrant (where both \( a \) and \( b \) are negative) point away from the positive axes, the actual angle is \( 225^\circ \).
This means the vector points further than halfway around the circle.

Vectors can lie in any of the four quadrants on a Cartesian coordinate plane. These quadrants determine specific characteristics of vectors' direction angles.
The third quadrant, where our vector \( \mathbf{u} = \langle -5, -5 \rangle \) is located, is defined by both x and y components being negative. This information alters how we interpret the direction angle.
For vectors in the third quadrant, the angle computed from tangent is adjusted by adding \( 180^\circ \) because they lie beyond the second quadrant.
Thus, for our vector, while the angle based on tangent is \( 45^\circ \), the actual direction angle is \( 225^\circ \) after adding these additional degrees.
This adjustment ensures the angle aligns with the vector's true path from the origin.

The tangent function is a fundamental trigonometric function that relates angles to side ratios in right triangles.
It's crucial in vector analysis for finding the direction angle, as it provides a relationship between the opposite (y) and adjacent (x) components.
The formula \( \tan \theta = \frac{b}{a} \) is used to derive \( \theta \). This equation tells us that \( \theta \) is the angle where its tangent equals the ratio of \( b \) (the y-component) to \( a \) (the x-component).
For example, in our vector \( \mathbf{u} = \langle -5, -5 \rangle \), the tangent function gives \( \tan^{-1}(1) = 45^\circ \). Yet, as previously noted, this must be corrected for the vector's position in the third quadrant, resulting in \( 225^\circ \).
The tangent function, thus, is instrumental in vector direction calculations, providing a bridge between algebraic expressions and geometric direction.