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


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Key Concepts

Vectors

A vector is a mathematical object that has both magnitude and direction, typically represented in a coordinate system. In two dimensions, a vector can be expressed in the form 〈a, b〉, where 'a' is the horizontal component and 'b' is the vertical component. Understanding how to manipulate and represent vectors is essential for solving problems in physics and engineering.

Magnitude of a Vector

The magnitude of a vector is a measure of its length and is calculated using the Pythagorean theorem. For a vector represented as 〈a, b〉, the magnitude is given by the formula √(a² + b²). This concept is crucial for understanding the size of a vector in relation to its direction and is often used in applications involving forces and motion.

Exact Values vs. Decimal Approximations

In mathematics, exact values refer to precise representations of numbers, such as fractions or radicals, while decimal approximations provide a rounded version of these values. When working with vectors, it is important to distinguish when to use exact values for clarity and precision, and when to round to a specified number of decimal places for practical applications, such as in engineering calculations.