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The dot product is a fundamental concept in vector calculus, often used to determine the relationship between two vectors. It provides a scalar result which is informative about how much one vector extends along another. For two vectors \( \vec{a} \) and \( \vec{b} \), the dot product is defined as:\[\vec{a} \cdot \vec{b} = \|\vec{a}\| \|\vec{b}\| \cos \theta\]where:

• \( \|\vec{a}\| \) and \( \|\vec{b}\| \) are the magnitudes of vectors \( \vec{a} \) and \( \vec{b} \).
• \( \theta \) is the angle between the two vectors.

In practical terms, for vectors given in coordinates, say \( \vec{a} = (a_1, a_2, ..., a_n) \) and \( \vec{b} = (b_1, b_2, ..., b_n) \), the dot product is calculated as:\[\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + ... + a_nb_n\]This operation is extremely useful when calculating the angle between two vectors, as seen in our problem with vectors \( \vec{w}_n \) and \( \vec{e}_k \).
The dot product formula \( \vec{w}_n \cdot \vec{e}_k = k \) helps us find the cosine of the angle, highlighting how a simple arithmetic operation can unveil deeper geometric insights.

The magnitude of a vector, sometimes referred to as the length or norm, conveys how long the vector is in Euclidean space. For any vector \( \vec{v} = (v_1, v_2, ..., v_n) \), its magnitude is calculated by:\[\| \vec{v} \| = \sqrt{v_1^2 + v_2^2 + ... + v_n^2}\]This formula simply reflects the Pythagorean theorem in higher dimensions. It's a critical component when determining the vector's size irrespective of its direction.

In the context of our exercise with the vector \( \vec{w}_n \), the magnitude is:\[\| \vec{w}_n \| = \sqrt{1^2 + 2^2 + ... + n^2}\]To simplify, this sum of squares can be expressed using the formula:\[\sum_{i=1}^n i^2 = \frac{n(n+1)(2n+1)}{6}\]This simplification gives us a neat computation of the length of \( \vec{w}_n \), helping in further calculations like finding angles or performing vector projections.

Basis vectors form the backbone of vector spaces. In simple terms, a basis set consists of vectors that are linearly independent and span the entire space. In the euclidean space \( \mathbb{R}^n \), the standard basis vectors are \( \vec{e}_1, \vec{e}_2, ..., \vec{e}_n \), where each vector has 1 in one coordinate and 0 in others.

• \( \vec{e}_1 = (1, 0, 0, ..., 0) \)
• \( \vec{e}_2 = (0, 1, 0, ..., 0) \)
• Continuing to \( \vec{e}_n = (0, 0, 0, ..., 1) \)

Each of these vectors essentially "points" in one coordinate direction, and any vector in \( \mathbb{R}^n \) can be constructed as a linear combination of them.

In the original problem, \( \vec{w}_n \) is a sum of scaled basis vectors. This highlights how they are utilized to build complex vectors in \( \mathbb{R}^n \). The use of basis vectors simplifies many operations since they are orthogonal, making dot products straightforward and reducing computational errors.

The angle between two vectors provides insight into their directional relationship. It's not just a geometric concept; it also dictates how much one vector projects onto another. If two vectors are orthogonal (perpendicular), the angle is 90°, and their dot product is zero.

For any vectors \( \vec{a} \) and \( \vec{b} \), the cosine of the angle \( \theta \) between them is given by:\[\cos \theta = \frac{\vec{a} \cdot \vec{b}}{\|\vec{a}\| \|\vec{b}\|}\]Here,

• \( \vec{a} \cdot \vec{b} \) denotes the dot product.
• \( \|\vec{a}\| \) and \( \|\vec{b}\| \) are the magnitudes of \( \vec{a} \) and \( \vec{b} \).

For our specific case in the problem, involving \( \vec{w}_n \) and \( \vec{e}_k \), the angle \( \alpha_{n,k} \) is expressed by the equation:\[\alpha_{n,k} = \cos^{-1}\left( \dfrac{k}{\| \vec{w}_n \|} \right)\]This demonstrates the elegance of vector calculus where even complex relationships can be distilled into manageable formulae. Over infinite limits, as given in the problem, this angle approaches \( \frac{\pi}{2} \) radians (or 90°), indicating that \( \vec{w}_n \) and \( \vec{e}_k \) become perpendicular.