### Understanding Vector Cross Product Components#### Problem ContextVector **u** has a **POSITIVE** x-component, **POSITIVE** y-component, and **ZERO** z-component. Vector **v** has a **NEGATIVE** x-component, **POSITIVE** y-component, and **POSITIVE** z-component.#### QuestionWhat is true about the cross product **u × v**?#### Answer Options- There is not enough information to know anything.- It has **NEGATIVE** x-component, **NEGATIVE** y-component, **NEGATIVE** z-component- It has **POSITIVE** x-component, **NEGATIVE** y-component, **NEGATIVE** z-component- It has **POSITIVE** x-component, **NEGATIVE** y-component, **POSITIVE** z-component#### Solution ExplanationTo determine the components of the cross product **u × v**, we can use the formula for the cross product of two vectors in three-dimensional space:Given\[ \mathbf{u} = (u_x, u_y, u_z) \]\[ \mathbf{v} = (v_x, v_y, v_z) \]The cross product **u × v** is given by:\[ \mathbf{u} \times \mathbf{v} = \left( u_y v_z - u_z v_y, u_z v_x - u_x v_z, u_x v_y - u_y v_x \right) \]Substitute the given values:\[ \mathbf{u} = (u_x > 0, u_y > 0, u_z = 0) \]\[ \mathbf{v} = (v_x < 0, v_y > 0, v_z > 0) \]So,\[ u \times v = \left( (u_y \cdot v_z - 0 \cdot v_y), (0 \cdot v_x - u_x \cdot v_z), (u_x \cdot v_y - u_y \cdot v_x) \right) \]\[ = (u_y \cdot v_z, -u_x \cdot v_z, u_x \cdot v_y - u_y \cdot v_x) \]### Interpreting the Signs:1. **u_y \cdot v_z**: - u_y > 0 (

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Vector u has POSITIVE x-component, POSITIVE y-component and ZERO z-component. Vector v has NEGATIVE x-component, POSITIVE y-component and POSITIVE z-component. What is true about u x v? There is not enough information to know anything. O It has NEGATIVE x-component, NEGATIVE y-component, NEGATIVE z-component O It has POSITIVE×-component, NEGATIVE y-component, NEGATIVE z-component It has POSITIVE x-component, NEGATIVE y-component, POSITIVE z-component

Vector u has POSITIVE x-component, POSITIVE y-component and ZERO z-component. Vector v has NEGATIVE x-component, POSITIVE y-component and POSITIVE z-component. What is true about u x v? There is not enough information to know anything. O It has NEGATIVE x-component, NEGATIVE y-component, NEGATIVE z-component O It has POSITIVE×-component, NEGATIVE y-component, NEGATIVE z-component It has POSITIVE x-component, NEGATIVE y-component, POSITIVE z-component

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.

Question

### Understanding Vector Cross Product Components

#### Problem Context
Vector **u** has a **POSITIVE** x-component, **POSITIVE** y-component, and **ZERO** z-component.
Vector **v** has a **NEGATIVE** x-component, **POSITIVE** y-component, and **POSITIVE** z-component.

#### Question
What is true about the cross product **u × v**?

#### Answer Options
- There is not enough information to know anything.
- It has **NEGATIVE** x-component, **NEGATIVE** y-component, **NEGATIVE** z-component
- It has **POSITIVE** x-component, **NEGATIVE** y-component, **NEGATIVE** z-component
- It has **POSITIVE** x-component, **NEGATIVE** y-component, **POSITIVE** z-component

#### Solution Explanation
To determine the components of the cross product **u × v**, we can use the formula for the cross product of two vectors in three-dimensional space:

Given
\[ \mathbf{u} = (u_x, u_y, u_z) \]
\[ \mathbf{v} = (v_x, v_y, v_z) \]

The cross product **u × v** is given by:
\[ \mathbf{u} \times \mathbf{v} = \left( u_y v_z - u_z v_y, u_z v_x - u_x v_z, u_x v_y - u_y v_x \right) \]

Substitute the given values:
\[ \mathbf{u} = (u_x > 0, u_y > 0, u_z = 0) \]
\[ \mathbf{v} = (v_x < 0, v_y > 0, v_z > 0) \]

So,
\[ u \times v = \left( (u_y \cdot v_z - 0 \cdot v_y), (0 \cdot v_x - u_x \cdot v_z), (u_x \cdot v_y - u_y \cdot v_x) \right) \]
\[ = (u_y \cdot v_z, -u_x \cdot v_z, u_x \cdot v_y - u_y \cdot v_x) \]

### Interpreting the Signs:
1. **u_y \cdot v_z**:
- u_y > 0 (

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