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Examples in Chapter 1

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Problem 1.42 Vector A has components Ax=1.30 cm and Ay= 2.25 cm; vector B has components Bx=4.10 cm, By=-3.75 cm. Find The components of the vector sum A+B the magnitude and direction of A+B The components of the vector B-A The magnitude and direction of B-A

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**The components of the vector sum A+B**

The sum of the x-components of A and B are equal to the x component of A+B

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**Find the magnitude of A+B**

(A+B)y (A+B)x (A+B)

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**Find the direction of A+B**

(A+B)y (A+B)x (A+B) q

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**Components, magnitude and direction of B-A**

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Problem 1.55

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**Your book’s way The book gives 3 formulae for C=AxB**

Cx=AyBz-AzBy Cy=AzBx-AxBz Cz=AxBy-AyBx Since A and B have only x- and y- components, we find Cz by Cz=4*(-2)-5*3=-8-15=-23 ||AxB||=23

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My way: First, a review? A determinant represents a single number and is used in linear algebra

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A 3 x 3 determinant

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For a cross-product

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For our problem

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Problem 1.61 Biological tissues are typically made up of 98% water. Given that the density of water is 1 x 103 kg/m3, estimate the mass of the heart of an adult human a cell with a diameter of 0.5 mm. a honey bee.

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Best guesses Human heart: size of fist (cylinder 4” long with diameter of 3”) Honey bee: 1” or 2.5 cm long cylinder with 0.25” diameter

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**Heart Problem 4” = 4*2.54 cm = 10.16 cm or 0.1016 m**

Volume of cylinder= p*(d/2)2 *L p * (7.62/2)2 *10.16=463 cm3 or cc Note: kg =1 pound

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**Book answer differs by order of magnitude**

A cell Assume spherical! Volume=4*p/3*(d/2)3 Book answer differs by order of magnitude

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Honey bee V=p*(d/2)2 *L=p*(.25/2)2 *1=0.049 in3 Book assumes ½ in long

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Problem 1.70 A sailor in a small sailboat encounters shifting winds. She sails 2.0 km east then 3.5 km southeast (-450 w.r.t. east) and then an unknown distance. Her final position is 5.8 km directly east of starting point. Find magnitude and direction of the third leg of the journey.

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Step 1: Draw it! 5.80 km 2.0 km 3.5 km

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**Step 2: Sketch in the details**

5.80 km 2.0 km 3.5 km Ay ?

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Step 3: Simplify 5.80 km-2.0 km=3.8km 2.0 km 3.5 km Ay Ay Ay ?

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**Step 4: Solve 5.80 km-2.0 km=3.8km 2.0 km Ay 3.5 km Ay Ay**

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**Step 5: 5.80 km-2.0 km=3.8km 2.0 km Ay 3.5 km Ay=2.47 km Ay**

?= =1.33 km=Ax

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Step 6: 1.33 km 2.47 km q

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