In Exercises 61–64, find the magnitude ||v||, to the nearest hundredth, and the direction angle θ, to the nearest tenth of a degree, for each given vector v. v = -10i + 15j

Key Concepts

Magnitude of a Vector

The magnitude of a vector represents its length and is calculated using the formula ||v|| = √(x² + y²), where x and y are the components of the vector. In this case, for the vector v = -10i + 15j, the magnitude can be found by substituting -10 for x and 15 for y, resulting in ||v|| = √((-10)² + (15)²).

Direction Angle of a Vector

The direction angle θ of a vector is the angle formed between the vector and the positive x-axis, measured counterclockwise. It can be calculated using the tangent function: θ = arctan(y/x). For the vector v = -10i + 15j, you would compute θ using the components -10 and 15, ensuring to consider the correct quadrant for the angle based on the signs of the components.

Quadrants in the Cartesian Plane

The Cartesian plane is divided into four quadrants, which are determined 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 which quadrant the vector lies in is crucial for accurately determining the direction angle θ, as it affects the angle's final value.