Find the angle θbetween each of the pairs of vectors u⃗ and...

Find the angle $\theta$ between each of the pairs of vectors $\vec{u}$ and $\vec{v}$.
$$\vec{u}=\left[\begin{array}{r}
1 \\
-1 \\
2 \\
-2
\end{array}\right], \vec{v}=\left[\begin{array}{l}
2 \\
3 \\
4 \\
5
\end{array}\right]$$

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Key Concepts

Dot Product

The dot product is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. It is calculated by multiplying corresponding entries and summing the results. The dot product is useful in determining the similarity in direction of two vectors and plays a crucial role in computing the angle between them.

Vector Magnitude

The magnitude (or length) of a vector is a measure of its size, determined by taking the square root of the sum of the squares of its components. It provides the scale of the vector and is essential when normalizing vectors or calculating the cosine of the angle between two vectors.

Cosine Formula for Calculating the Angle Between Vectors

The cosine formula relates the dot product of two vectors to their magnitudes and the cosine of the angle between them, expressed as cos? = (u · v) / (||u|| ||v||). This formula allows for the determination of the angle between two vectors by first computing both the dot product and the magnitudes of each vector.

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