In Exercises 47–52, write the vector v in terms of i and j whose magnitude ||v|| and direction angle θ are given. ||v|| = 1/2, θ = 113°

Key Concepts

Magnitude of a Vector

The magnitude of a vector represents its length or size, denoted as ||v||. In this case, the magnitude is given as 1/2, indicating that the vector's length is half a unit. Understanding magnitude is essential for determining how far the vector extends in space.

Direction Angle

The direction angle θ of a vector indicates the angle it makes with the positive x-axis, measured in degrees or radians. Here, θ is 113°, which means the vector is oriented 113° counterclockwise from the positive x-axis. This angle is crucial for calculating the vector's components along the x and y axes.

Vector Components

Vector components are the projections of a vector along the coordinate axes, typically represented as i (x-component) and j (y-component). To express the vector v in terms of i and j, we use the formulas v = ||v|| * cos(θ) * i + ||v|| * sin(θ) * j. This allows us to break down the vector into its horizontal and vertical parts based on its magnitude and direction angle.