91Ó°ÊÓ

Vectors are fundamental in physics and mathematics, representing quantities that have both direction and magnitude. The magnitude, often denoted as \(||\bm{v}||\), represents the length or size of the vector.
In general, for any vector \(\bm{v} = \\), the magnitude is calculated using the formula:
\[ ||\bm{v}|| = \sqrt{a^2 + b^2} \]
This is derived from applying the Pythagorean theorem to the vector, treating it as the hypotenuse of a right triangle with legs 'a' and 'b'.
For example, consider the vector \<0, -6\>.
Let's calculate its magnitude: \[ ||\bm{v}|| = \sqrt{0^2 + (-6)^2} = \sqrt{36} = 6 \]
Therefore, the magnitude of the vector \<0, -6\> is 6. This value indicates how long the vector is, regardless of its direction.

The direction angle of a vector tells us the angle the vector makes with the positive x-axis. This angle provides a clear indication of the vector’s orientation in a plane.
To find the direction angle \(\theta\) of a vector \, we use the tangent function, which relates the angle to the ratio of the y-component to the x-component of the vector:
\[ \tan \theta = \frac{b}{a} \]
For the vector \<0, -6\>, this calculation becomes:
\[ \tan \theta = \frac{-6}{0} \]
Since division by zero is undefined, \(\tan \theta\) is undefined, indicating the vector lies directly on one of the axes.
In this case, the vector lies along the negative y-axis. Therefore, the direction angle \(\theta\) is 270° (or equivalently -90°), representing the vector pointing straight down.

The tangent function is crucial for finding angles in trigonometry, particularly in the context of vectors.
This function, defined as the ratio of the opposite side to the adjacent side in a right-angled triangle, is given by \[ \tan \theta = \frac{b}{a} \]
Here, 'b' is the y-component and 'a' is the x-component of the vector.
The tangent function helps us determine the direction angle of vectors by revealing the ratio of these components.
For example, with the vector \<0, -6\>, when we calculated \[ \tan \theta = \frac{-6}{0} \]
Since this value is undefined, it indicates the vector points directly along one of the coordinate axes.
Understanding the tangent function helps us properly interpret the direction and orientation of vectors in a plane and is vital for solving problems involving vector angles.