Find the magnitude and direction angle for each vector. Round angle measures to the nearest tenth, as necessary.
〈-4, -7〉

Verified step by step guidance

1. Identify the components of the vector. Here, the vector is given as \(\langle -4, -7 \rangle\), where \(x = -4\) and \(y = -7\).
2. Calculate the magnitude of the vector using the formula \(\text{magnitude} = \sqrt{x^2 + y^2}\). Substitute the values to get \(\sqrt{(-4)^2 + (-7)^2}\).
3. Find the direction angle \(\theta\) of the vector relative to the positive x-axis using the formula \(\theta = \tan^{-1}\left(\frac{y}{x}\right)\). Substitute the values to get \(\theta = \tan^{-1}\left(\frac{-7}{-4}\right)\).
4. Since both \(x\) and \(y\) are negative, the vector lies in the third quadrant. Adjust the angle found in the previous step by adding \(180^\circ\) to get the correct direction angle.
5. Round the direction angle to the nearest tenth of a degree as required.

Key Concepts

Vector Magnitude

The magnitude of a vector 〈x, y〉 represents its length and is calculated using the Pythagorean theorem as √(x² + y²). This gives a non-negative scalar value indicating how long the vector is regardless of its direction.

Direction Angle of a Vector

The direction angle of a vector is the angle it makes with the positive x-axis, measured counterclockwise. It can be found using the arctangent function: θ = arctan(y/x), with adjustments based on the vector's quadrant to get the correct angle.

Quadrant Considerations for Angles

Since arctan(y/x) only returns values between -90° and 90°, the vector's quadrant must be considered to determine the correct direction angle. For vectors in the third quadrant (both x and y negative), add 180° to the arctan value to find the proper angle.