https://file.aiquickdraw.com/mathbot/file/1726381807252lhlm7bq7.jpg

Here are the solutions to the problems:

a. \( \langle \mathbf{u}, \mathbf{v} \rangle = 12 \)

b. \( \langle k\mathbf{v}, \mathbf{w} \rangle = -18 \)

c. \( \langle \mathbf{u} + \mathbf{v}, \mathbf{w} \rangle = -9 \)

d. \( \|\mathbf{v}\| = \sqrt{30} \)

e. \( d(\mathbf{u}, \mathbf{v}) = \|\mathbf{u} - \mathbf{v}\| = \sqrt{11} \)

f. \( \|\mathbf{u} - k\mathbf{v}\| = \sqrt{203} \)

Let me know if you need further details on any of these, and here are 5 related questions for practice:

1. How do you compute the inner product of two vectors in a standard Euclidean space?
2. What changes when we apply a weighted Euclidean inner product compared to the standard one?
3. What is the geometric interpretation of \( \|\mathbf{v}\| \) in a vector space?
4. How does scalar multiplication of a vector affect its norm?
5. How do you calculate the distance between two vectors in \( \mathbb{R}^2 \)?

**Tip**: When working with inner products, pay attention to any weights or changes in the metric, as they affect the overall calculation.

Let R^2 have the weighted Euclidean inner product ⟨u,v⟩ = 2u₁v₁ + 3u₂v₂, and let u = (1,1), v = (3,2), w = (0,−1), and k = 3. Compute the following: a. ⟨u,v⟩ b. ⟨kv,w⟩ c. ⟨u+v,w⟩ d. ||v|| e. d(u,v) f. ||u−kv||

Learn how to calculate weighted Euclidean inner products, vector norms, and distances in R². Solve problems involving u = (1,1), v = (3,2), w = (0,−1), and scalar multiplication with k = 3. This problem is suitable for undergraduate-level students studying linear algebra and vector spaces.

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