Write the vector v in terms of i and j whose magnitude ||v|| and direction angle θ are given.


||v|| = 10, θ = 330°

Verified step by step guidance

1. Step 1: Understand that a vector \( \mathbf{v} \) in the plane can be expressed in terms of its magnitude and direction angle using the formula \( \mathbf{v} = ||\mathbf{v}|| \cdot (\cos \theta \mathbf{i} + \sin \theta \mathbf{j}) \).
2. Step 2: Substitute the given magnitude \( ||\mathbf{v}|| = 10 \) and direction angle \( \theta = 330^\circ \) into the formula.
3. Step 3: Calculate \( \cos 330^\circ \) and \( \sin 330^\circ \). Recall that \( 330^\circ \) is in the fourth quadrant where cosine is positive and sine is negative.
4. Step 4: Use the values from Step 3 to express the vector \( \mathbf{v} \) as \( 10 \cdot (\cos 330^\circ \mathbf{i} + \sin 330^\circ \mathbf{j}) \).
5. Step 5: Simplify the expression to write \( \mathbf{v} \) in terms of \( \mathbf{i} \) and \( \mathbf{j} \).

Key Concepts

Vector Representation

A vector in a two-dimensional space can be represented in terms of its components along the x-axis and y-axis, typically denoted as 'i' and 'j'. The vector v can be expressed as v = ai + bj, where 'a' and 'b' are the respective components. Understanding this representation is crucial for converting magnitude and direction into component form.

Magnitude of a Vector

The magnitude of a vector is a measure of its length and is denoted by ||v||. It can be calculated using the formula ||v|| = √(a² + b²), where 'a' and 'b' are the components of the vector. In this question, the magnitude is given as 10, which will be used to find the components of the vector.

Direction Angle

The direction angle θ of a vector is the angle formed with the positive x-axis, measured counterclockwise. In this case, θ = 330° indicates that the vector is positioned in the fourth quadrant. The angle is used to determine the components of the vector using trigonometric functions: a = ||v|| cos(θ) and b = ||v|| sin(θ).