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A unit vector is a vector with a magnitude of exactly 1. It serves as a building block for constructing other vectors in any direction. Although its direction can vary, its length always remains the same—1 unit. This property makes unit vectors very useful for defining direction without affecting magnitude in vector operations.

• Denoted often with a hat symbol, for example, \(\mathbf{u}\hat{}\).
• Main property: \(|\mathbf{u}| = 1\).
• Used to simplify calculations and standardize magnitude.

Recognizing a unit vector is crucial when working with dot products since it simplifies the calculation by setting one of the magnitudes to 1. In problems involving angles, as in our example, knowing that \(\mathbf{u}\) is a unit vector ensures that you're multiplying by 1, making calculations easier.

The angle between two vectors is a measure of their direction relative to each other. It ranges from 0 radians (or degrees) when vectors point in exactly the same direction, up to \(\pi\) radians (180 degrees) when they point in opposite directions.

• An angle of \(0\) indicates parallel vectors.
• An angle of \(\pi\) indicates antiparallel vectors.

In the dot product formula \(\mathbf{u} \cdot \mathbf{v} = |\mathbf{u}| \cdot |\mathbf{v}| \cdot \cos (\theta)\), the cosine of the angle tells us about the directional relationship:

• \(\cos(\theta)\) is positive when vectors point roughly in the same direction; negative when they point oppositely.

For the given problem, the angle \(\theta\) is \(\frac{3\pi}{4}\), which is more than \(\pi/2\), leading to \(\cos(\theta)\) being negative, indicating opposite directional relations.

The magnitude of a vector refers to its length or size. It's a scalar value that quantifies how much of the vector there is; think of it as the distance from the head to the tail of the arrow symbol that represents a vector.

• Notation: usually denoted \(|\mathbf{v}|\).
• Cannot be negative; it is always zero or a positive number.

Computing magnitude is straightforward with simple operations. For a vector \( \mathbf{v} = (x, y) \), the magnitude is calculated using the Pythagorean theorem: \(|\mathbf{v}| = \sqrt{x^2 + y^2}\) for 2D vectors, or \(|\mathbf{v}| = \sqrt{x^2 + y^2 + z^2}\) for 3D.
In the provided example, \(|\mathbf{v}|\) is given as 2. This information is crucial for determining the dot product, because the dot product scales with the magnitude of each vector, weighted by the directional relationship expressed through \(\cos(\theta)\).