Need help with this practice problem 3 **Problem 3:** The cross product is rather esoteric on its own, so let's see how we can interpret what it actually means.(a) Given two vectors $\vec{a}$ and $\vec{b}$, prove that $|\vec{a} \times \vec{b}|$ is the area of the parallelogram spanned by the two. Note: this relation gives rise to the notion of an **area vector**, which will be especially relevant to PHY 9C.(b) According to the picture of integral calculus, if we were to stretch this spanned parallelogram along some arbitrary third vector $\vec{c}$, we would get a 3D parallelepiped. What quantity would then be the volume of the parallelepiped? Does this comply with what would happen to the parallelepiped if $\vec{c}$ was in the same plane with $\vec{a}$ and $\vec{b}$? **Diagram:** A red arrow is shown stretching from a point on a plane upwards to indicate the vector $\vec{c}$ extending the parallelogram into a 3D parallelepiped.(c) What does the sign of the quantity you found in (b) signify?---In this problem, the primary focus is to understand the geometrical interpretation of the cross product and its relation to physical quantities such as area and volume.1. **Area of Parallelogram:** Derive that the magnitude of the cross product $|\vec{a} \times \vec{b}|$ gives the area of the parallelogram formed by vectors $\vec{a}$ and $\vec{b}$.2. **Volume of Parallelepiped:** Extend this concept to three dimensions by considering a third vector $\vec{c}$ and show that the volume of the resulting parallelepiped is given by the scalar triple product $\vec{a} \cdot (\vec{b} \times \vec{c})$. Additionally, explore the implications when $\vec{c}$ lies in the same plane as $\vec{a}$ and $\vec{b}$, which should theoretically yield a zero volume.3. **Significance of Sign:** Discuss the significance of the sign of the scalar triple product in terms of orientation and right-hand rule.

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Description: Need help with this practice problem 3 **Problem 3:** The cross product is rather esoteric on its own, so let's see how we can interpret what it actually means.(a) Given two vectors $\vec{a}$ and $\vec{b}$, prove that $|\vec{a} \times \vec{b}|$

Given that parallelogram is we know that area of parallelogram is…

Math

Calculus

Problem 3: The cross product is rather esoteric on its own, so let's see how we can interpret what it actually means. (a) Given two vectors ā and 5, prove that jā x b| is the area of the parallelogram spanned by the two. Note: this relation gives rise to the notion of an area vector, which will be especially relevant to PHY 9C. (b) According to the picture of integral calculus, if we were to stretch this spanned parallelogram along some arbitrary third vector č, we would get a 3D parallelepiped. What quantity would then be the volume of the parallelepiped? Does this comply with what would happen to the parallelepiped if was in the same plane with ä and b? (c) What does the sign of the quatity you found in (b) signify?

Problem 3: The cross product is rather esoteric on its own, so let's see how we can interpret what it actually means. (a) Given two vectors ā and 5, prove that jā x b| is the area of the parallelogram spanned by the two. Note: this relation gives rise to the notion of an area vector, which will be especially relevant to PHY 9C. (b) According to the picture of integral calculus, if we were to stretch this spanned parallelogram along some arbitrary third vector č, we would get a 3D parallelepiped. What quantity would then be the volume of the parallelepiped? Does this comply with what would happen to the parallelepiped if was in the same plane with ä and b? (c) What does the sign of the quatity you found in (b) signify?

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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Binomial Distribution

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Need help with this practice problem 3

**Problem 3:** The cross product is rather esoteric on its own, so let's see how we can interpret what it actually means.

(a) Given two vectors $\vec{a}$ and $\vec{b}$, prove that $|\vec{a} \times \vec{b}|$ is the area of the parallelogram spanned by the two.
Note: this relation gives rise to the notion of an **area vector**, which will be especially relevant to PHY 9C.

(b) According to the picture of integral calculus, if we were to stretch this spanned parallelogram along some arbitrary third vector $\vec{c}$, we would get a 3D parallelepiped. What quantity would then be the volume of the parallelepiped? Does this comply with what would happen to the parallelepiped if $\vec{c}$ was in the same plane with $\vec{a}$ and $\vec{b}$?

**Diagram:**
A red arrow is shown stretching from a point on a plane upwards to indicate the vector $\vec{c}$ extending the parallelogram into a 3D parallelepiped.

(c) What does the sign of the quantity you found in (b) signify?

---

In this problem, the primary focus is to understand the geometrical interpretation of the cross product and its relation to physical quantities such as area and volume.

1. **Area of Parallelogram:** Derive that the magnitude of the cross product $|\vec{a} \times \vec{b}|$ gives the area of the parallelogram formed by vectors $\vec{a}$ and $\vec{b}$.

2. **Volume of Parallelepiped:** Extend this concept to three dimensions by considering a third vector $\vec{c}$ and show that the volume of the resulting parallelepiped is given by the scalar triple product $\vec{a} \cdot (\vec{b} \times \vec{c})$. Additionally, explore the implications when $\vec{c}$ lies in the same plane as $\vec{a}$ and $\vec{b}$, which should theoretically yield a zero volume.

3. **Significance of Sign:** Discuss the significance of the sign of the scalar triple product in terms of orientation and right-hand rule.

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