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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 thatu脳v=0.

Step by step solution

01

Step 1. Given Information

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

02

Step 2. Theorem 10.26 

Theorem states that, "The cross product of two parallel vectors u and v in 鈩�3is u脳v=0."

03

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 csuch that v=cu. Consider their cross product:

role="math" localid="1649915602184" u脳v=u脳(cu)u脳v=c脳(u脳u)u脳v=c脳0u脳v=0

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Most popular questions from this chapter

Give an example of three vectors in 鈩�3that form a right-handed triple. Explain how you can use the same three vectors to form a left-handed triple.

If u and v are nonzero vectors in 鈩�3, why do the equations role="math" localid="1649263352081" u路(u脳v)=0 and v路(u脳v)=0 tell us that the cross product is orthogonal to both u and v?

In Exercises 24-27, find compuv,projuv and the component of v orthogonal tou.

role="math" localid="1649693816584" u=3,1,-2,v=-6,-2,4

In Exercises 22鈥�29 compute the indicated quantities when u=(2,1,鈭�3),v=(4,0,1),andw=(鈭�2,6,5)

Find the area of the parallelogram determined by the vectors u and v.

Sketch the parallelogram determined by the two vectors (1,2)and (3,鈭�1). How can you use the cross product to find the area of this parallelogram?
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