Question

Describe what each code snippet does and how it might be used in Statics. Describe in general terms what Statics calculation is being performed by each code snippet. What is input to the calculation and what results are given? Give definitions not numbers. Hint: all arrays are components of vectors [ x, y, z ].

Code 1

V1 = [ 0, 1, 2 ];
V2 = [ 3, -4, 5 ];
x = 1;
y = 2;
z = 3;
vXv = [ V1(y)*V2(z) - V1(z)*V2(y), V1(z)*V2(x) - V1(x)*V2(z) ...
    , V1(x)*V2(y) - V1(y)*V2(x)];

Note: ... must be at the end of the line as it is the Matlab way to continue a statement on the next line.

Code 2

V1 = [ 0, 1, 2 ];
V2 = [ 3, -4, 5 ];
x = 1;
y = 2;
z = 3;
vDv = V1(x)*V2(x) + V1(y)*V2(y) + V1(z)*V2(z)

Code 3:

V = [ 6, -2, 3 ];
Mag = sqrt( sum(V .^2 ));
lambda = V ./ Mag

Code 4

v = [ 1, 2, 3 ];
r1 = [ 0.3, -0.4, 0.5 ];
r2 = r1 + v;
r1Xf = cross(r1,v);
r2Xf = cross(r2,v);
isequal(r1Xf,r2Xf)

What should be the result of this snippet?

Next, write a section of code as described below:

Write a code to compute all 6 permutations of the mixed triple product of the three vectors:

P→=0i→+3j→+0k→

Q→=0i→+0j→+5k→

S→=2i→+0j→+0k→

Then change the first vector to,  P→=2i→+3j→+5k→ and compare the results. To make this easier type the commands "help dot" and "help cross" into Matlab.

Answer 1

A code to compute all 6 permutations of the mixed triple product of the three vectors is given as follows:

Code 1:

V1 = [0,1,2];
Declares a vector 0i + 1j + 2k
V2 = [3,-4,5];
Declares a vector 3i - 4j + 5k
x = 1;
Stores a value of 1 in x
y = 2;
?Stores a value of 2 in y
z = 3;
?Stores a value of 3 in z
vXv = [V1(y)*V2(z) - V1(z)*V2(y), V1(z)*V2(x) - V1(x)*V2(z), V1(x)*V2(y) - V1(y)*V2(x)] ;
Calculates the cross product if the two vectors V1 and V2
vXv = [13 6 -3] is the MATLAB output.

Code 2:

V1 = [0,1,2];
Declares a vector 0i + 1j + 2k
V2 = [3,-4,5];
Declares a vector 3i - 4j + 5k
x = 1;
Stores a value of 1 in x
y = 2;
?Stores a value of 2 in y
z = 3;
?Stores a value of 3 in z
vDv = V1(x)*V2(x) + V1(y)*V2(y) + V1(z)*V2(z);
Calculates the dot product if the two vectors V1 and V2
vDv = 6 is the MATLAB output.

Code 3:

V = [6,-2,3];
Declares a vector 6i - 2j + 3k
Mag = sqrt (sum(V.^2));
V.^2 does the aquare of each element and hence V.^2 = [36,4,9]
sum(V.^2) does the sum of all elements of V.^2 and hence sum = 36+4+9 = 49
sqrt (sum(V.^2)) gives the square root of sum(V.^2) and hence sqrt (sum(V.^2)) = 7
So, M = 7
lamda = V ./ Mag;
V ./ Mag divides each element of V by Mag
So, lamda = is the MATLAB output

Code 4:

v = [1,2,3];
Declares a vector 1i + 2j + 3k
r1 = [0.3, -0.4, 0.5];
Declares a vector 0.3i - 0.4j + 0.5k
r2 = r1 + v;
Element-wise addition of the vectors r1 and v occurs and the result get stores in r2
r2 = [1.3 1.6 3.5]
r1Xf = cross(r1,v);
r1Xf stores the value of the cross product of r1 and v, r1 X v.
r1Xf = [-2.2 -0.4 1]
r2Xf = cross(r2,v);
r2Xf stores the value of the cross product of r2 and v, r2 X v.
r2Xf = [-2.2 -0.4 1]
isequal(r1Xf ,r2Xf)
The MATLAB output is 0 since r1Xf and r2Xf are both equal.

The MATLAB code is as follows:

%Declaring Vector P
P = [0, 3, 0];
%Declaring Vector Q
Q = [0, 0, 5];
%Declaring Vector S
S = [2, 0, 0];
%Finding the first mixed triple product P.(Q X S)
A1 = dot(P,cross(Q,S))
%A1 = 30
%Finding the second mixed triple product P.(S X Q)
B1 = dot(P,cross(S,Q))
%B1 = -30
%Finding the third mixed triple product Q.(P X S)
C1 = dot(Q,cross(P,S))
%C1 = 30
%Finding the fouth mixed triple product Q.(S X P)
D1 = dot(Q,cross(S,P))
%D1 = 30
%Finding the fifth mixed triple product S.(P X Q)
E1 = dot(S,cross(P,Q))
%E1 = 30
%Finding the sixth mixed triple product S.(Q X P)
F1 = dot(S,cross(Q,P))
%F1 = 30
%Redcalaring the vector P to the new value
P = [2, 3, 5];
%Finding the first mixed triple product P.(Q X S)
A2 = dot(P,cross(Q,S))
%A2 = 30
%Finding the second mixed triple product P.(S X Q)
B2 = dot(P,cross(S,Q))
%B2 = -30
%Finding the third mixed triple product Q.(P X S)
C2 = dot(Q,cross(P,S))
%C2 = -30
%Finding the fouth mixed triple product Q.(S X P)
D2 = dot(Q,cross(S,P))
%D2 = 30
%Finding the fifth mixed triple product S.(P X Q)
E2 = dot(S,cross(P,Q))
%E2 = 30
%Finding the sixth mixed triple product S.(Q X P)
F2 = dot(S,cross(Q,P))
%F2 = -30
% The corresponding values of the mixed triple product remain unchanged since the new P = old P + Q + S., i.e. A1 = A2, B1 = B2, C1 = C2, D1 = D2, E1 = E2, F1 = F2