### Vector Analysis Problem Set**(1) Let**\[\mathbf{u} = \begin{pmatrix} -1 \\ 2 \end{pmatrix} \quad \text{and} \quad \mathbf{v} = \begin{pmatrix} 4 \\ 2 \end{pmatrix}.\]**Questions:**(a) Are **u** and **v** orthogonal?(b) What is the distance between **u** and **v**?(c) Find a unit vector in the direction of **u**.---### ExplanationIn this problem, we are given two vectors in two-dimensional space and are asked to perform several operations:1. **Orthogonality Check:** - Two vectors are orthogonal if their dot product is zero. Compute \(\mathbf{u} \cdot \mathbf{v}\).2. **Distance Calculation:** - The distance between two vectors is given by the Euclidean distance formula: \(\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).3. **Unit Vector:** - A unit vector in the direction of \(\mathbf{u}\) is found by dividing each component of \(\mathbf{u}\) by its magnitude: \(\|\mathbf{u}\|\). The magnitude of a vector \(\mathbf{u} = \begin{pmatrix} x \\ y \end{pmatrix}\) is \(\sqrt{x^2 + y^2}\).Use these principles to solve the given tasks related to vectors \(\mathbf{u}\) and \(\mathbf{v}\).

Answered: Image /qna-images/answer/d8de2924-e894-466d-8a11-f6e218d47e40.jpg

Answered: -1 4 and v = 2 u = 2 (a) Are u and v…

-1 4 and v = 2 u = 2 (a) Are u and v orthogonal? (b) What is the distance between u and v? (c) Find a unit vector in the direction of u. (;)-

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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