Chapter 10: Q2MP (page 535)

Let $e_{1}, e_{2}, e_{3}$ bea set of orthogonal unit vectors forming a right-handed system if taken in cyclic order. Show that the triple scalarproduct . $e_{i} . (e_{j} \times e_{k}) = ε_{i j k}$

Short Answer

Expert verified

It has been shown that the scalar triple product is . $e_{i} \cdot (e_{j} \times e_{k}) = ε_{i j k}$

Step by step solution

Given Information

The given set of orthogonal unit vectors is . $e_{1}, e_{2}, e_{3}$

Definition of a cartesian tensor

The first rank tensor is just a vector. A tensor of the second rank has nine components (in three dimensions) in every rectangular coordinate system.

Find the value

Let $i^{t h}$ component is given by. $i^{t h} {(a \times b)}_{i} = \sum_{l m} ε_{i l m} a_{l} b_{m}$

The scalar triple product is given below.

role="math" localid="1664357511514" $\begin{array}{l} e_{i} \cdot (e_{j} \times e_{k}) = {(e_{j} \times e_{k})}_{i} \\ e_{i} \cdot (e_{j} \times e_{k}) = \sum_{l m} ε_{i l m} δ_{j l} δ_{k m} \\ e_{i} \cdot (e_{j} \times e_{k}) = ε_{i j k} \end{array}$

Hence,thescalar triple product is . $e_{i} \cdot (e_{j} \times e_{k}) = ε_{i j k} \begin{array}{l} \end{array}$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Let $e_{1}, e_{2}, e_{3}$ bea set of orthogonal unit vectors forming a right-handed system if taken in cyclic order. Show that the triple scalarproduct . $e_{i} . (e_{j} \times e_{k}) = ε_{i j k}$

Short Answer

Step by step solution

Given Information

Definition of a cartesian tensor

Find the value

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Translational Dynamics

Force

Mechanics and Materials

Geometrical and Physical Optics

Atoms and Radioactivity

Oscillations

Study anywhere. Anytime. Across all devices.

Let e1,e2,e3bea set of orthogonal unit vectors forming a right-handed system if taken in cyclic order. Show that the triple scalarproduct .ei.(ej×ek)=εijk

Short Answer

Step by step solution

Given Information

Definition of a cartesian tensor

Find the value

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Translational Dynamics

Force

Mechanics and Materials

Geometrical and Physical Optics

Atoms and Radioactivity

Oscillations

Study anywhere. Anytime. Across all devices.

Let $e_{1}, e_{2}, e_{3}$ bea set of orthogonal unit vectors forming a right-handed system if taken in cyclic order. Show that the triple scalarproduct . $e_{i} . (e_{j} \times e_{k}) = ε_{i j k}$