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

Verified step by step guidance

1. Step 1: To find the magnitude of the vector \( \langle -7, 24 \rangle \), use the formula for the magnitude of a vector \( \langle a, b \rangle \), which is \( \sqrt{a^2 + b^2} \). Substitute \( a = -7 \) and \( b = 24 \) into the formula.
2. Step 2: Calculate the magnitude by evaluating \( \sqrt{(-7)^2 + 24^2} \). This involves squaring each component of the vector, adding the results, and then taking the square root of the sum.
3. Step 3: To find the direction angle \( \theta \) of the vector, use the formula \( \theta = \tan^{-1}\left(\frac{b}{a}\right) \). Substitute \( a = -7 \) and \( b = 24 \) into the formula.
4. Step 4: Calculate \( \theta = \tan^{-1}\left(\frac{24}{-7}\right) \). This will give you the angle in radians or degrees, depending on your calculator settings.
5. Step 5: Since the vector \( \langle -7, 24 \rangle \) is in the second quadrant (negative x-component and positive y-component), adjust the angle obtained from the inverse tangent function to ensure it is in the correct quadrant. Add 180 degrees to the angle if necessary to find the correct direction angle.

Key Concepts

Magnitude of a Vector

The magnitude of a vector is a measure of its length or size, calculated using the formula √(x² + y²), where x and y are the vector's components. For the vector 〈-7, 24〉, the magnitude would be √((-7)² + (24)²) = √(49 + 576) = √625 = 25.

Direction Angle of a Vector

The direction angle of a vector is the angle formed between the vector and the positive x-axis, typically measured in degrees. It can be found using the tangent function: θ = arctan(y/x). For the vector 〈-7, 24〉, the angle would be θ = arctan(24/-7), which requires consideration of the vector's quadrant to determine the correct angle.

Quadrants in the Coordinate Plane

The coordinate plane is divided into four quadrants, each defined by the signs of the x and y coordinates. The first quadrant has both coordinates positive, the second has a negative x and positive y, the third has both negative, and the fourth has a positive x and negative y. Understanding the quadrant is essential for correctly interpreting the direction angle, especially when the vector has a negative x-component.