**Vector Analysis and Calculations**In this exercise, we evaluate vectors **u** and **v**:- **u** = $(0, 1, \sqrt{7})$- **v** = $(-1, \sqrt{7}, -1)$### Tasks:(a) **Find the Norms of u and v**- $\| \mathbf{u} \| = 2.8284$- $\| \mathbf{v} \| = 3$(b) **A Unit Vector in the Direction of v**- Unit vector: $\left(-\frac{1}{3}, \frac{\sqrt{7}}{3}, -\frac{1}{3}\right)$(c) **A Unit Vector in the Direction Opposite That of u**- Incorrect attempt: $\left(0, \frac{\sqrt{7}}{\sqrt{8}}, -\frac{1}{\sqrt{8}}\right)$(d) **Dot Product of u and v**- $\mathbf{u} \cdot \mathbf{v} = 0$(e) **Dot Product of u with Itself**- $\mathbf{u} \cdot \mathbf{u} = 8$(f) **Dot Product of v with Itself**- $\mathbf{v} \cdot \mathbf{v} = 9$### Explanation:- **Norm of a Vector**: The length or magnitude of the vector, calculated using the square root of the sum of the squares of its components. - **Unit Vector**: A vector with magnitude 1 in the same direction as the given vector. Calculated by dividing each component of the vector by its magnitude.- **Dot Product**: A scalar product that represents the sum of the products of the corresponding entries of two sequences of numbers.This exercise involves linear algebra concepts relevant in physics, engineering, and computer graphics for analyzing vector directions and interactions.

Since you have asked a question with multiple question in a single request, we would be answering…

Use a software program or a graphing utility with vector capabilities to find the lengths of u and v, a unit vector in the direction of v, a unit vector in the direction opposite that of u, u· v, u · u, and v· v. (Round your answers to four decimal places.) - (), 1, v7). v-(-1, v7, -1) (a) the norms of u and v |u|| - 2.82843 |v|| - 3 (b) a unit vector in the direction of v V7 3 (c) a unit vector in the direction opposite that of u 0, V8

Use a software program or a graphing utility with vector capabilities to find the lengths of u and v, a unit vector in the direction of v, a unit vector in the direction opposite that of u, u· v, u · u, and v· v. (Round your answers to four decimal places.) - (), 1, v7). v-(-1, v7, -1) (a) the norms of u and v |u|| - 2.82843 |v|| - 3 (b) a unit vector in the direction of v V7 3 (c) a unit vector in the direction opposite that of u 0, V8

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Question

**Vector Analysis and Calculations**

In this exercise, we evaluate vectors **u** and **v**:

- **u** = $(0, 1, \sqrt{7})$
- **v** = $(-1, \sqrt{7}, -1)$

### Tasks:

(a) **Find the Norms of u and v**
- $\| \mathbf{u} \| = 2.8284$
- $\| \mathbf{v} \| = 3$

(b) **A Unit Vector in the Direction of v**
- Unit vector: $\left(-\frac{1}{3}, \frac{\sqrt{7}}{3}, -\frac{1}{3}\right)$

(c) **A Unit Vector in the Direction Opposite That of u**
- Incorrect attempt: $\left(0, \frac{\sqrt{7}}{\sqrt{8}}, -\frac{1}{\sqrt{8}}\right)$

(d) **Dot Product of u and v**
- $\mathbf{u} \cdot \mathbf{v} = 0$

(e) **Dot Product of u with Itself**
- $\mathbf{u} \cdot \mathbf{u} = 8$

(f) **Dot Product of v with Itself**
- $\mathbf{v} \cdot \mathbf{v} = 9$

### Explanation:

- **Norm of a Vector**: The length or magnitude of the vector, calculated using the square root of the sum of the squares of its components.

- **Unit Vector**: A vector with magnitude 1 in the same direction as the given vector. Calculated by dividing each component of the vector by its magnitude.

- **Dot Product**: A scalar product that represents the sum of the products of the corresponding entries of two sequences of numbers.

This exercise involves linear algebra concepts relevant in physics, engineering, and computer graphics for analyzing vector directions and interactions.

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