Question

Matlab Questions (please show what code I need, including any comments, for each of the questions):

1)  Initialize two row vectors a and b of arbitrary but equal lengths (pick any length ³3) with random numbers between 0 and 1. Recalling that the dot product between two vectors is a ∙ b = |a||b| cos theta, calculate the angle, theta, between the two vectors (radians or degrees is fine, just state what you used).

Note: • You can use built-in inverse trigonometric functions from Matlab • You may not use the built-in “dot” and “norm” functions from Matlab • You should not write products like a(1)*b(1)+a(2)*b(2)+…, etc.

2) (Extra credit) Consider vectors a=[1 1 1] and b=[1 -1 1]. These two linearly independent vectors define a plane. Calculate the unit normal to this plane. You can use any Matlab functions you like.

3) There are 100 runners in the Amherst 5k Hot Chocolate race; let’s label them by jersey numbers 1 to 100. To simulate this race, make a 1D array consisting of 100 entries with random finishing times that are uniformly distributed between 20 and 30 minutes. Each entry in this array corresponds to the finishing time for each runner, e.g., entry #29 is the finishing time for runner 29. Using only array operations and/or array functions: a. Print the top three finishing times; your output could look something like this: Top three finishing times: 20.2094 21.6666 21.8455 b. Print the top three runners; your output coould look something like this: Top three runners: 62 37 88 [Hint: zyBooks Sec 6.7] c. If 25 minutes is the qualifying time for the Hampshire county 5k race, use logical operators to calculate how many runners qualified for the county race.

4) You are asked to design a water storage tank with a capacity of 1,000 liters. The tank will be fabricated from stainless steel (rolled sheet), which cost $100/ sq. m. You are asked to consider three possible designs for the tank: Option 1 – a cube; Option 2 – a cylinder; Option 3 – a sphere. Calculate the cost of each of these designs and report the least expensive option.

• You should create a local function that has two inputs: tank capacity (volume) and material cost per sq. m. • Your function should evaluate and report the cost of each option ordered from lowest to highest. The output does not have to be fancy; something like this is adequate: Options: 3 1 2 Pricing ($): 90.8400 100.9100 123.4600 • For the cylinder, you can assume that the height and radius are equal (1 liter = 103 cm3 = 10-3 m3 )

Answer 1

clc
clear
close
%(1)
a=rand(1,3); % assigning random number to a
b=rand(1,3); % assigning random number to b
theta=acos(sum(a.*b)/sqrt(sum(a.^2)*sum(b.^2))) %theta in radian
%(2)
a=[1 1 1]; b=[1 -1 1];
UNv=cross(a,b)/sqrt(norm(a)^2*norm(b)^2-norm(a.*b)^2)