Find the magnitude and direction angle for each vector. Round angle measures to the nearest tenth, as necessary.
〈5, 7〉

Verified step by step guidance

1. Identify the components of the vector \( \langle 5, 7 \rangle \), where 5 is the horizontal component \( x \) and 7 is the vertical component \( y \).
2. Calculate the magnitude of the vector using the formula \( \sqrt{x^2 + y^2} \). Substitute \( x = 5 \) and \( y = 7 \) into the formula.
3. Determine the direction angle \( \theta \) using the formula \( \theta = \tan^{-1}\left(\frac{y}{x}\right) \). Substitute \( x = 5 \) and \( y = 7 \) into the formula.
4. Convert the angle from radians to degrees if necessary, and round the angle measure to the nearest tenth.
5. Ensure the angle is in the correct quadrant based on the signs of the vector components. Since both components are positive, the angle is in the first quadrant.

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 〈5, 7〉, the magnitude represents the distance from the origin to the point (5, 7) in a Cartesian coordinate system.

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, where the angle θ is calculated as θ = arctan(y/x). For the vector 〈5, 7〉, this angle indicates the vector's orientation in the plane.

Rounding Angles

Rounding angles is the process of adjusting the angle measure to a specified degree of precision, often to the nearest tenth. This is important in trigonometry to provide clear and concise answers, especially when dealing with angles that may not be whole numbers. For example, if the calculated angle is 53.13 degrees, rounding to the nearest tenth would yield 53.1 degrees.