Chapter 1: Q2P (page 4)

Is the cross product associative?
$(A×B)×C=A×(B×C)$
If so, prove it; if not, provide a counterexample (the simpler the better).

Short Answer

Expert verified

The values of the two cross product $A \times (B \times C)$ and $(A \times B) \times C$ , are not equal. Thus $(A \times B) \times C \neq A \times (B \times C)$ .

Step by step solution

Explain the concept and describe the given information

The cross product is associative if the result of cross products $A×(B×C)$ and $(A×B)×C$ is same.

Assume the vectors

To prove, let us assume three vectors as $A = 4 i + 3 j, B = 3 i + 2 j$ , and $C = 6 i + 5 j$ .

Obtain the value of $A \times (B \times C)$ using the vectors assumed.

$A \times (B \times C) = (4 i + 3 j) \times (3 i + 2 j) \times (6 i + 5 j) = (4 i + 3 j) \times (15 k - 12 k) = (4 i + 3 j) \times (3 k) = (12 j + 9 i) . . . . . . . . . (1)$

Obtain the value of $(A \times B) \times C$ using the vectors assumed.

role="math" localid="1653052563790" $(A \times B) \times C = ((4 i + 3 j) \times 3 i + 2 j)) \times (6 i + 5 j) = (12 k - 9 k) \times (6 i + 5 j) = (3 k) \times (6 i + 5 j) = 18 j - 15 i . . . . . . (2)$

Compare the values.

It can be concluded from the equation (1) and (2), that the values of the two cross product $A \times (B \times C)$ and $(A \times B) \times C$ , are not equal. Thus

$(A \times B) \times C \neq A \times (B \times C)$

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Is the cross product associative?
$(A×B)×C=A×(B×C)$
If so, prove it; if not, provide a counterexample (the simpler the better).

Short Answer

Step by step solution

Explain the concept and describe the given information

Assume the vectors

Compare the values.

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Is the cross product associative? (A×B)×C=A×(B×C)If so, prove it; if not, provide a counterexample (the simpler the better).

Short Answer

Step by step solution

Explain the concept and describe the given information

Assume the vectors

Compare the values.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Mechanics and Materials

Physics of Motion

Astrophysics

Further Mechanics and Thermal Physics

Atoms and Radioactivity

Turning Points in Physics

Study anywhere. Anytime. Across all devices.

Is the cross product associative?
$(A×B)×C=A×(B×C)$
If so, prove it; if not, provide a counterexample (the simpler the better).