Write the vector v in terms of i and j whose magnitude ||v|| and direction angle θ are given.
||v|| = 8, θ = 45°

Key Concepts

Magnitude of a Vector

The magnitude of a vector represents its length or size, denoted as ||v||. It is a scalar quantity that can be calculated using the Pythagorean theorem in two dimensions, where ||v|| = √(v_x² + v_y²). In this case, the magnitude is given as 8, indicating the vector's overall length.

Direction Angle

The direction angle θ of a vector is the angle formed between the vector and the positive x-axis, measured in degrees or radians. It provides information about the vector's orientation in the coordinate system. For this problem, θ = 45° indicates that the vector is oriented equally between the x and y axes.

Vector Representation in Component Form

A vector can be expressed in component form using unit vectors i and j, where i represents the x-component and j represents the y-component. The components can be calculated using the formulas v_x = ||v|| * cos(θ) and v_y = ||v|| * sin(θ). For the given magnitude and angle, this allows us to express the vector v as v = v_x * i + v_y * j.