Find the magnitude and direction angle for each vector. Round angle measures to the nearest tenth, as necessary. See Example 1.
〈8√2, -8√2〉

Verified step by step guidance

1. Step 1: Identify the components of the vector. The vector is given as \( \langle 8\sqrt{2}, -8\sqrt{2} \rangle \), where \( a = 8\sqrt{2} \) and \( b = -8\sqrt{2} \).
2. Step 2: Calculate the magnitude of the vector using the formula \( \| \mathbf{v} \| = \sqrt{a^2 + b^2} \). Substitute the values of \( a \) and \( b \) into the formula.
3. Step 3: Simplify the expression under the square root to find the magnitude. This involves squaring each component, adding them, and then taking the square root.
4. Step 4: Determine the direction angle \( \theta \) using the formula \( \theta = \tan^{-1}\left(\frac{b}{a}\right) \). Substitute the values of \( a \) and \( b \) into the formula.
5. Step 5: Adjust the angle \( \theta \) based on the quadrant in which the vector lies. Since \( a > 0 \) and \( b < 0 \), the vector is in the fourth quadrant, so ensure the angle is measured correctly in standard position.

Key Concepts

Magnitude of a Vector

The magnitude of a vector represents its length and is calculated using the formula √(x² + y²), where x and y are the vector's components. For the vector 〈8√2, -8√2〉, the magnitude can be found by substituting the values into this formula, providing a measure of how far the vector extends from the origin.

Direction Angle of a Vector

The direction angle of a vector is the angle it makes with the positive x-axis, typically measured in degrees. It can be calculated using the arctangent function: θ = arctan(y/x). For the vector 〈8√2, -8√2〉, this involves determining the angle based on the ratio of the y-component to the x-component, taking into account the quadrant in which the vector lies.

Quadrants in the Coordinate Plane

The coordinate plane is divided into four quadrants, each defined by the signs of the x and y coordinates. Understanding which quadrant a vector lies in is crucial for determining the correct direction angle. For instance, the vector 〈8√2, -8√2〉 is in the fourth quadrant, where x is positive and y is negative, affecting the angle's final value.