Chapter 10: Q. 65 (page 825)

Use the definition of the cross product to prove that the cross product of two parallel vectors is $0$ . (This is Theorem 10.26.)

Short Answer

Expert verified

Hence, we prove that $u \times v = 0$ .

Step by step solution

Step 1. Given Information

Use the definition of the cross product to prove that the cross product of two parallel vectors is $0$ .

Step 2. Theorem 10.26

Theorem states that, "The cross product of two parallel vectors u and v in $ℝ^{3}$ is $u \times v = 0$ ."

Step 3. As we know that the cross product of a vector with itself is 0.

Let $u$ and v be parallel vectors, let there be a scalar $c$ such that $v = c u$ . Consider their cross product:

role="math" localid="1649915602184" $u \times v = u \times (c u) u \times v = c \times (u \times u) u \times v = c \times 0 u \times v = 0$

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Use the definition of the cross product to prove that the cross product of two parallel vectors is $0$ . (This is Theorem 10.26.)

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Theorem 10.26

Step 3. As we know that the cross product of a vector with itself is 0.

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Use the definition of the cross product to prove that the cross product of two parallel vectors is 0. (This is Theorem 10.26.)

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Theorem 10.26

Step 3. As we know that the cross product of a vector with itself is 0.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Decision Maths

Discrete Mathematics

Theoretical and Mathematical Physics

Applied Mathematics

Logic and Functions

Study anywhere. Anytime. Across all devices.

Use the definition of the cross product to prove that the cross product of two parallel vectors is $0$ . (This is Theorem 10.26.)