Examples in Chapter 1

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

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

Find the magnitude of A+B (A+B)y (A+B)x (A+B)

Find the direction of A+B (A+B)y (A+B)x (A+B) q

Components, magnitude and direction of B-A

Problem 1.55

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

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

A 3 x 3 determinant

For a cross-product

For our problem

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.

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

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: 0.454 kg =1 pound

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

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

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.

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

Step 2: Sketch in the details 5.80 km 2.0 km 3.5 km Ay ?

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

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

Step 5: 5.80 km-2.0 km=3.8km 2.0 km Ay 3.5 km Ay=2.47 km Ay ?=3.8-2.47=1.33 km=Ax

Step 6: 1.33 km 2.47 km q