91Ó°ÊÓ

When dealing with vectors, one important concept is magnitude. The magnitude of a vector is essentially its length and can be visualized as the distance from the vector's start point to its end point. For a vector \(\textbf{v} = \langle x, y \rangle\), the formula to calculate magnitude is \(\/|\mathbf{v}\/| = \sqrt{x^2 + y^2}\).

Let's break this down further:

• The components of the vector represent its coordinates in space.
• To determine the vector's magnitude, we apply the Pythagorean theorem.
• We square each component, sum them, and take the square root of the total.

For example, in the given exercise, the vector \(\textbf{A} - \textbf{B}\) has components \(\textbf{A} - \textbf{B} = \langle 5, -2\rangle\). So the magnitude is calculated as:

\(|\mathbf{A} - \mathbf{B}\/| \ = \sqrt{5^2 + (-2)^2} = \sqrt{25 + 4} = \sqrt{29}\).

This value represents the length of the vector \(\textbf{A} - \textbf{B}\).

The direction angle of a vector indicates the angle it makes with the positive x-axis. This angle is essential for understanding the vector's orientation in the plane.

To find this angle, we use the inverse tangent function. For a vector \(\textbf{v} = \langle x, y \rangle\), the direction angle \(\theta\) is given by the formula: \(\theta = \tan^{-1}\bigg(\frac{y}{x}\bigg)\).

Here are the steps involved:

• Extract the components of the vector.
• Divide the y-component by the x-component.
• Apply the inverse tangent function (usually available on a scientific calculator).

In our exercise, for \(\textbf{A} - \textbf{B} = \langle 5, -2 \rangle\), the direction angle is: \(\theta = \tan^{-1}\bigg(\frac{-2}{5}\bigg)\).

When calculated, \(\theta\) approximately equals -21.8°, which means the vector points below the x-axis, indicating a downward direction.

The tangent function relates to right-angle triangles and is fundamental in trigonometry. It's particularly useful in finding angles and relating the sides of a right triangle.

For a right-angle triangle, the tangent of an angle \(\theta\) is defined as the ratio of the opposite side to the adjacent side:

\(\tan(\theta) = \frac{opposite}{adjacent}\).

When it comes to vectors:

• The y-component is considered the opposite side.
• The x-component is the adjacent side.
• Using these, we can find the direction angle as discussed.

Let's reiterate using our example: The vector \(\textbf{A} - \textbf{B} = \langle 5, -2 \rangle\) has opposite side -2 and adjacent side 5. The direction angle is:

\(\theta = \tan^{-1}\bigg(\frac{-2}{5}\bigg)\).

This fundamental function helps in converting a vector's components to an understandable angle, clarifying its direction.