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Chapter 11: Q. 40 (page 860)

Evaluate and simplify the indicated quantities in Exercises 35–41.
$((\sin t) i + (\cos t) j + t k) \times ((\cos t) i + (\sin t) j)$

Short Answer

Expert verified

The simplification of $((\sin t) i + (\cos t) j + t k) \times ((\cos t) i + (\sin t) j)$ is $(- t \sin t, t \cos t, \sin^{2} t - \cos^{2} t)$ .

Step by step solution

01

Step 1. Given Information.

The quantities:

$((\sin t) i + (\cos t) j + t k) \times ((\cos t) i + (\sin t) j)$

02

Step 2. Do the cross product.

By Cross product,

$((\sin t) i + (\cos t) j + t k) \times ((\cos t) i + (\sin t) j) = ((\cos t) (0) - (\sin t) t, (\cos t) t - (\sin t) 0, (\sin t) (\sin t) - (\cos t) (\cos t)) = (- t \sin t, t \cos t, \sin^{2} t - \cos^{2} t)$

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Most popular questions from this chapter

$Complete the following definition : If r (t) = <x (t), y (t), z (t)> is a differentiable position function, then the velocity vector v (t) is . . . .$

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Let k be a scalar and $r (t)$ be a differentiable vector function. Prove that $\frac{d}{d t} (k r (t)) = k r' (t)$ . (This is Theorem 11.11 (a).)

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Annie is conscious of tidal currents when she is sea kayaking. This activity can be tricky in an area south-southwest of Cattle Point on San Juan Island in Washington State. Annie is planning a trip through that area and finds that the velocity of the current changes with time and can be expressed by the vector function

$<0.4 \cos (\frac{π (t - 8)}{6}), - 1.1 \cos (\frac{π (t - 11)}{6})>,$

where t is measured in hours after midnight, speeds are given in knots and point due north.

(a) What is the velocity of the current at 8:00 a.m.?

(b) What is the velocity of the current at 11:00 a.m.?

(c) Annie needs to paddle through here heading southeast, 135 degrees from north. She wants the current to push her. What is the best time for her to pass this point? (Hint: Find the dot product of the given vector function with a vector in the direction of Annie’s travel, and determine when the result is maximized.)

See all solutions

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