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

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Identify the given vectors from the image and note their magnitudes and directions (angles) if provided.

Recall that a vector in the form \( \langle a, b \rangle \) can be expressed using its magnitude \( r \) and direction angle \( \theta \) as \( \langle r \cos(\theta), r \sin(\theta) \rangle \).

For each vector, calculate the \( a \) component by multiplying the magnitude by \( \cos(\theta) \), i.e., \( a = r \cos(\theta) \).

Similarly, calculate the \( b \) component by multiplying the magnitude by \( \sin(\theta) \), i.e., \( b = r \sin(\theta) \).

Write each vector in the form \( \langle a, b \rangle \) using exact values (like \( \sqrt{3}/2 \)) or decimal approximations rounded to four decimal places as appropriate.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Vector Representation in Component Form

Vectors can be expressed as ordered pairs 〈a, b〉, where 'a' and 'b' represent the vector's horizontal and vertical components, respectively. This form simplifies vector operations like addition, subtraction, and scalar multiplication by working with components directly.

Trigonometric Functions for Component Calculation

To find vector components from magnitude and direction, use trigonometric functions: the horizontal component is magnitude × cos(θ), and the vertical component is magnitude × sin(θ), where θ is the angle the vector makes with the positive x-axis.

Exact Values and Decimal Approximations

When expressing vector components, exact values involve using known trigonometric ratios (like √2/2), while decimal approximations round these values to a specified number of decimal places, such as four, to balance precision and readability.