91Ó°ÊÓ

To find the magnitude of a vector, we use the distance formula from geometry, which is derived from the Pythagorean theorem. A vector \(\textbf{v}\) in a 2D plane typically has components \(x\) and \(y\). Its magnitude, denoted as \(||\textbf{v}||\), can be calculated using the formula: \[ ||\textbf{v}|| = \sqrt{x^2 + y^2} \].
This formula gives the length of the vector, which can be thought of as the straight-line distance from the origin to the point \( (x, y) \).
In our example, the vector \( \textbf{B} - \textbf{A} \) is \(\textbf{v} = \langle -5, 2 \rangle\).
Plugging in the components into our formula:
\[|| \textbf{B} - \textbf{A} || = \sqrt{(-5)^2 + 2^2} = \sqrt{25 + 4} = \sqrt{29} \]. So, the magnitude of \( \textbf{B} - \textbf{A} \) is \( \sqrt{29}\).

The direction angle of a vector represents the angle between the positive x-axis and the vector.
To find this angle, we use the arctangent function (also known as inverse tangent).
For a vector \( \textbf{v} = \langle x, y \rangle \), the direction angle \( \theta \) is calculated using: \[ \theta = \tan^{-1}\left(\frac{y}{x} \right) \].
Remember, if the vector lies in the second or third quadrant, adjustments are needed for the correct direction angle. In our step-by-step solution, we have the vector \( \textbf{B} - \textbf{A} = \langle -5, 2 \rangle \).
The formula becomes: \[ \theta = \tan^{-1}\left(\frac{2}{-5}\right) \].
This gives an angle of approximately \(-21.8^\text{°}\).
Since the vector is in the second quadrant (negative x and positive y), we add \(\text{180°}\) to the angle:
\[\theta = -21.8^\text{°} + 180^\text{°} = 158.2^\text{°} \].
Therefore, the direction angle is \(158.2^\text{°}\).

Trigonometry and vectors are closely tied together. Trigonometric functions are essential for finding vector magnitudes and direction angles.
When dealing with vectors, often you'll encounter the need to decompose or resolve vectors into their components using sine and cosine functions. For instance, given an angle \(\theta\) and a magnitude \(||\textbf{v}||\), the x and y components can be found using:
\(\textbf{v} \cos(\theta)\) for the x-component.
\(\textbf{v} \sin(\theta)\) for the y-component.
This way, any vector can be broken down and analyzed using trigonometric principles, making it easier to work with in different contexts, such as physics or engineering.
In summary, trigonometry helps us understand vectors by providing the tools to compute angles, directions, and lengths in a straightforward manner.