In Exercises 22–24, sketch each vector as a position vector and find its magnitude. v = -3i - 4j

Key Concepts

Position Vectors

A position vector represents a point in space relative to an origin. In a two-dimensional Cartesian coordinate system, it is expressed in terms of its components along the x-axis and y-axis, typically denoted as v = xi + yj, where x and y are the coordinates. For the vector v = -3i - 4j, the position vector indicates a point located at (-3, -4) in the plane.

Magnitude of a Vector

The magnitude of a vector is a measure of its length and is calculated using the Pythagorean theorem. For a vector v = xi + yj, the magnitude is given by the formula |v| = √(x² + y²). In the case of v = -3i - 4j, the magnitude would be |v| = √((-3)² + (-4)²) = √(9 + 16) = √25 = 5.

Vector Components

Vector components break down a vector into its individual parts along the coordinate axes. In the vector v = -3i - 4j, the component -3 corresponds to the x-direction (horizontal) and -4 to the y-direction (vertical). Understanding these components is essential for visualizing the vector's direction and calculating its magnitude.