   Chapter 13.2, Problem 2PT

Chapter
Section
Textbook Problem

True or False: [ r ( t ) × s ( t ) ] ′ = r ( t ) × s ′ ( t ) + s ( t ) × r ′ ( t ) .

To determine

Whether the given statement “[r(t)×s(t)]r(t)×s(t)+s(t)×r(t)” is true or false.

Explanation

The given statement is, “[r(t)×s(t)]=r(t)×s(t)+r(t)×s(t)”.

Let r(t)=f1(t)i+g1(t)j+h1(t)k and s(t)=f2(t)i+g2(t)j+h2(t)k then r(t)=f1(t)i+g1(t)j+h1(t)k and s(t)=f2(t)i+g2(t)j+h2(t)k.

Obtain the cross product of the vector functions r(t) and s(t).

r(t)×s(t)=|ijkf1(t)g1(t)h1(t)f2(t)g2(t)h2(t)|={(g1(t)h2(t)g2(t)h1(t))i(f1(t)h2(t)f2(t)h1(t))j+(f1(t)g2(t)f2(t)g1(t))k}

Differentiate the above vector function with respect to t.

[r(t)×s(t)]={ddt(g1(t)h2(t)g2(t)h1(t))iddt(f1(t)h2(t)f2(t)h1(t))j+ddt(f1(t)g2(t)f2(t)g1(t))k}={(ddt(g1(t)h2(t))ddt(g2(t)h1(t)))i(ddt(f1(t)h2(t))ddt(f2(t)h1(t)))j+(ddt(f1(t)g2(t))ddt(f2(t)g1(t)))k}={(g1(t)h2(t)+g1(t)h2(t)(g2(t)h1(t)+g2(t)h1(t)))i((f1(t)h2(t)+f1(t)h2(t))(f2(t)h1(t)+f2(t)h1(t)))j+(f1(t)g2(t)+f1(t)g2(t)(f2(t)g1(t)+f2(t)g1(t)))k}

={(g1(t)h2(t)+g1(t)h2(t)g2(t)h1(t)g2(t)h1(t))i+(f2(t)h1(t)+f2(t)h1(t)f1(t)h2(t)f1(t)h2(t))j+(f1(t)g2(t)+f1(t)g2(t)f2(t)g1(t)f2(t)g1(t))k} (1)

Obtain the cross product of the vector functions r(t) and s(t).

r(t)×s(t)=|ijkf1(t)g1(t)h1(t)f2(t)g2(t)h2(t)|={(g1(t)h2(t)g2(t)h1(t))i(f1(t)h2(t)f2(t)h1(t))j+(f1(t)g2(t)f2(t)g1(t))k}

={(g1(t)h2(t)g2(t)h1(t))i(+f2(t)h1(t)f1(t)h2(t))j+(f1(t)g2(t)f2(t)g1(t))k}                                                          (2)

Similarly, obtain the cross product of the vector functions r(t) and s(t)

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