Write each vector in the form 〈a, b〉. Write answers using exact values or to four decimal places, as appropriate.

Verified step by step guidance

1. Identify the given vectors from the image and note their magnitudes and directions (angles) if provided.
2. Recall that a vector in the form \( \langle a, b \rangle \) can be expressed using its magnitude \( r \) and angle \( \theta \) as \( \langle r \cos(\theta), r \sin(\theta) \rangle \).
3. For each vector, calculate the \( a \) component by multiplying the magnitude by \( \cos(\theta) \), i.e., \( a = r \cos(\theta) \).
4. Similarly, calculate the \( b \) component by multiplying the magnitude by \( \sin(\theta) \), i.e., \( b = r \sin(\theta) \).
5. Write each vector in the form \( \langle a, b \rangle \) using exact values (like fractions or square roots) or decimal approximations rounded to four decimal places as appropriate.

Key Concepts

Vector Representation in Component Form

Vectors in the plane can be expressed as ordered pairs 〈a, b〉, where 'a' and 'b' represent the horizontal (x) and vertical (y) components, respectively. This form allows for easy manipulation and calculation of vector operations such as addition, subtraction, and scalar multiplication.

Trigonometric Functions for Component Calculation

To find the components of a vector given its magnitude and direction, use trigonometric functions: the x-component is magnitude × cos(θ), and the y-component is magnitude × sin(θ), where θ is the angle the vector makes with the positive x-axis. This method converts polar form to rectangular form.

Exact Values and Decimal Approximations

When expressing vector components, exact values involve using known trigonometric values (like √2/2 for 45°), while decimal approximations round these values to a specified number of decimal places (e.g., four). Choosing between exact and approximate depends on the problem's requirements for precision.