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

Verified step by step guidance

1. Recall that a vector \( \mathbf{v} \) in two dimensions can be expressed in terms of the unit vectors \( \mathbf{i} \) and \( \mathbf{j} \) as \( \mathbf{v} = v_x \mathbf{i} + v_y \mathbf{j} \), where \( v_x \) and \( v_y \) are the components of the vector along the x-axis and y-axis respectively.
2. Use the magnitude \( ||\mathbf{v}|| = 8 \) and the direction angle \( \theta = 45^\circ \) to find the components. The components can be found using the formulas: \( v_x = ||\mathbf{v}|| \cos(\theta) \) and \( v_y = ||\mathbf{v}|| \sin(\theta) \).
3. Substitute the given values into the component formulas: \( v_x = 8 \cos(45^\circ) \) and \( v_y = 8 \sin(45^\circ) \).
4. Calculate the cosine and sine of \( 45^\circ \) (you can use the exact values or a calculator), then multiply by 8 to find the components \( v_x \) and \( v_y \).
5. Write the vector \( \mathbf{v} \) in terms of \( \mathbf{i} \) and \( \mathbf{j} \) as \( \mathbf{v} = v_x \mathbf{i} + v_y \mathbf{j} \) using the components found in the previous step.

Key Concepts

Vector Representation in the Plane

A vector in two dimensions can be expressed as a combination of unit vectors i and j along the x and y axes, respectively. Writing a vector in terms of i and j involves finding its horizontal and vertical components, which correspond to the vector's projections on these axes.

Magnitude and Direction of a Vector

The magnitude of a vector represents its length, while the direction angle θ indicates the angle it makes with the positive x-axis. These two parameters uniquely define the vector's position in the plane and are essential for converting between polar and rectangular forms.

Using Trigonometric Functions to Find Components

The horizontal (x) and vertical (y) components of a vector can be found using cosine and sine of the direction angle θ, respectively. Specifically, x = ||v|| cos θ and y = ||v|| sin θ, which allows expressing the vector as v = x i + y j.