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 . $e_{i} \cdot (e_{j} \times e_{k}) = ε_{i j k} \begin{array}{l} \end{array}$

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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

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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

Astrophysics

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Classical Mechanics

Atoms and Radioactivity

Particle Model of Matter

Fields in Physics

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 . $e_{i} . (e_{j} \times e_{k}) = ε_{i j k}$