Explanation Given: Let B denotes s binomial vector at a point on a space curve C, defined by B = T × N . Formula Used: B = T × N T ( t ) = r ' ( t ) ‖ r ' ( t ) ‖ N ( t ) = N ' ( t ) ‖ N ' ( t ) ‖ T ( t ) = r ' ( t ) ‖ r ' ( t ) ‖ Proof: Take 1 ‖ r ' ( t ) ‖ = a ( t ) T ( t ) = a ( t ) r ' ( t ) Differentiating, T ' ( t ) = a ( t ) r ' ' ( t ) + a ' ( t ) + r ' ( t ) Substitute this value in N ( t ) = N ' ( t ) ‖ N ' ( t ) ‖ , we get N ( t ) = a ( t ) r ' ' ( t ) + a ' ( t ) + r ' ( t ) ‖ N ' ( t ) ‖ Now, take ‖ N ' ( t ) ‖ = b ( t ) , we get N ( t ) = a ( t ) r

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Chapter 14.4, Problem 91E

To determine

To prove:

$r' \times r''$ is a multiple of B. Also, conclude that $B = \frac{r' \times r''}{‖ r' \times r'' ‖}$ .

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Chapter 14.4, Problem 90E

Chapter 14.4, Problem 92E

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Chapter 14 Solutions

CALCULUS 4E (LL) W/ ACHIEVE ACCESS

Ch. 14.1 - Prob. 1PQ Ch. 14.1 - Prob. 2PQ Ch. 14.1 - Prob. 3PQ Ch. 14.1 - Prob. 4PQ Ch. 14.1 - Prob. 5PQ Ch. 14.1 - Prob. 6PQ Ch. 14.1 - Prob. 1E Ch. 14.1 - Prob. 2E Ch. 14.1 - Prob. 3E Ch. 14.1 - Prob. 4E

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To prove: r ' × r ' ' is a multiple of B. Also, conclude that B = r ' × r ' ' ‖ r ' × r ' ' ‖ .

Solution Summary: The author proves that r is a multiple of B. B = T N T ( t ) = s binomial vector at point on space curve C.

Videos

To prove:

$r' \times r''$ is a multiple of B. Also, conclude that $B = \frac{r' \times r''}{‖ r' \times r'' ‖}$ .

Want to see the full answer?

Chapter 14 Solutions

To prove: r ' × r ' ' is a multiple of B. Also, conclude that B = r ' × r ' ' ‖ r ' × r ' ' ‖ .

Solution Summary: The author proves that r is a multiple of B. B = T N T ( t ) = s binomial vector at point on space curve C.

Videos

To prove: r'×r'' is a multiple of B. Also, conclude that B=r'×r''‖r'×r''‖.

Want to see the full answer?

Chapter 14 Solutions

To prove:

$r' \times r''$ is a multiple of B. Also, conclude that $B = \frac{r' \times r''}{‖ r' \times r'' ‖}$ .