Chapter 10: Q. 66 (page 825)

Use the definition of the cross product to prove that the cross product of two vectors u and v is anti-commutative; that is, prove that $u \times v = - v \times u$ . (This is
Theorem 10.27.)

Short Answer

Expert verified

Hence, we prove that $u \times v = - (v \times u)$ .

Step by step solution

Step 1. Given Information

Use the definition of the cross product to prove that the cross product of two vectors u and v is anti-commutative; that is, prove that $u \times v = - v \times u$ . (This is Theorem 10.27.)

Step 2. Theorem 10.27

Theorem states that "For any vectors u and v in $ℝ^{3}$ $u \times v = - (v \times u)$ ."

Step 3. We have to prove u×v=−(v×u)

Firstly finding the value of $u \times v$

role="math" localid="1649916954076"

u \times v = (u_{1}, u_{2}, u_{3}) \times (v_{1}, v_{2}, v_{3}) u \times v = [\begin{matrix} i & j & k \\ u_{1} & u_{2} & u_{3} \\ v_{1} & v_{2} & v_{3} \end{matrix}] u \times v = i |\begin{matrix} u_{2} & u_{3} \\ v_{2} & v_{3} \end{matrix}| - j |\begin{matrix} u_{1} & u_{3} \\ v_{1} & v_{3} \end{matrix}| + k |\begin{matrix} u_{1} & u_{2} \\ v_{1} & v_{2} \end{matrix}| u \times v = i (u_{2} v_{3} - u_{3} v_{2}) - j (u_{1} v_{3} - u_{3} v_{1}) + k (u_{1} v_{2} - u_{2} v_{1}) u \times v = (u_{2} v_{3} - u_{3} v_{2}, - u_{1} v_{3} + u_{3} v_{1}, u_{1} v_{2} - u_{2} v_{1})

Step 4. Now finding the value of v×u

$v \times u = (u_{1}, u_{2}, u_{3}) \times (v_{1}, v_{2}, v_{3}) v \times u = [\begin{matrix} i & j & k \\ v_{1} & v_{2} & v_{3} \\ u_{1} & u_{2} & u_{3} \end{matrix}] v \times u = i |\begin{matrix} v_{2} & v_{3} \\ u_{2} & u_{3} \end{matrix}| - j |\begin{matrix} v_{1} & v_{3} \\ u_{1} & u_{3} \end{matrix}| + k |\begin{matrix} v_{1} & v_{2} \\ u_{1} & u_{2} \end{matrix}| v \times u = i (v_{2} u_{3} - v_{3} u_{2}) - j (v_{1} u_{3} - v_{3} u_{1}) + k (v_{1} u_{2} - v_{2} u_{1}) v \times u = (v_{2} u_{3} - v_{3} u_{2}, - v_{1} u_{3} + v_{3} u_{1}, v_{1} u_{2} - v_{2} u_{1}) v \times u = - (u_{3} v_{2} - u_{2} v_{3}, - u_{3} v_{1} + u_{1} v_{3}, u_{2} v_{1} - u_{1} v_{2})$

Hence it shows that $u \times v = - (v \times u)$

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Use the definition of the cross product to prove that the cross product of two vectors u and v is anti-commutative; that is, prove that $u \times v = - v \times u$ . (This is
Theorem 10.27.)

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Theorem 10.27

Step 3. We have to prove u×v=−(v×u)

Step 4. Now finding the value of v×u

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Use the definition of the cross product to prove that the cross product of two vectors u and v is anti-commutative; that is, prove that u×v=−v×u. (This isTheorem 10.27.)

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Theorem 10.27

Step 3. We have to prove u×v=−(v×u)

Step 4. Now finding the value of v×u

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Theoretical and Mathematical Physics

Probability and Statistics

Geometry

Calculus

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Use the definition of the cross product to prove that the cross product of two vectors u and v is anti-commutative; that is, prove that $u \times v = - v \times u$ . (This is
Theorem 10.27.)