Find the magnitude and direction angle for each vector. Round angle measures to the nearest tenth, as necessary. See Example 1.
〈-7, 24〉

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Identify the components of the vector given as \( \langle -7, 24 \rangle \), where \( x = -7 \) and \( y = 24 \).

Calculate the magnitude of the vector using the formula \( \text{magnitude} = \sqrt{x^2 + y^2} \). Substitute the values to get \( \sqrt{(-7)^2 + 24^2} \).

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 \( \theta = \tan^{-1} \left( \frac{24}{-7} \right) \).

Since the x-component is negative and the y-component is positive, the vector lies in the second quadrant. Adjust the angle \( \theta \) accordingly by adding 180 degrees if necessary to find the correct direction angle.

Round the direction angle to the nearest tenth of a degree as required.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Vector Magnitude

The magnitude of a vector represents its length and is calculated using the Pythagorean theorem. For a vector 〈x, y〉, the magnitude is √(x² + y²). This gives a non-negative scalar value indicating the vector's size regardless of 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 inverse tangent function: θ = arctan(y/x). Adjustments may be needed based on the vector's quadrant to get the correct angle.

Quadrant Considerations in Angle Calculation

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 quadrants II, III, or IV, add 180° or 360° as needed to place the angle in the correct range (0° to 360°).

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