#Name: Irina Mukhametzhanova
#This homework is OK to post


#12.1 Homework
#Exercise 5:
#X-component of u = cos(45 degrees) * ||u|| = (sqrt(2)/2)||u||
#Y-component of u = sin(45 degrees) * ||u|| = (sqrt(2)/2)||u||

#Exercise 7:
#X-component of w = cos(20 degrees) * ||w|| ≈ 0.94||w||
#Y-component of w = -sin(20 degrees) * ||w|| ≈ -0.342||w||

#Exercise 9:
P:= (3,2);
Q:= (2,7);
V1:= [Q-P];
#Result: V1 := [-1,5]

#Exercise 11:
P:= (3,5);
Q:= (1,-4);
V1:= [Q-P];
#Result: V1 := [-2,-9]

#Exercise 15:
V1:= 5 * [6,2];
#Result: V1 := [30, 10]

#Exercise 21:
#2v = [2,3]*2 = [4,6]
#-w = [4,1]*-1 = [-4,-1]
#v+w = [2,3] + [4,1] = [6,4]
#2v-w = [4,6] - [4,1] = [0,5]
#The sketch is attached as an image in the email

#Exercise 41:
Magnitude := (V) -> sqrt(V[1]^2 + V[2]^2);
Magnitude([3,4]);
#Result: 5
U := ([3, 4])/Magnitude([3, 4]);
#Result: U := [3/5, 4/5]

#Exercise 47:
#X-component of e = cos(4pi/7)*||e|| = cos(4pi/7) ≈ -0.22
#Y-component of e = sin(4pi/7)*||e|| = sin(4pi/7) ≈ 0.97
#Resulting vector: [-0.22, 0.97]


#12.2 Homework
#Exercise 11:
R:= [1,4,3];
W:= [3,-2,3];
P:= R - W;
#Result: P := [-2, 6, 0]

#Exercise 13:
#a)
#v is 4 times this vector - so, they are parallel, and point to the same direction
#b)
#v is not a multiple of this vector - so, they are not parallel, and they do not point in the same direction
#c)
#v is (-4/7) times this vector - so, they are parallel. However, they point in opposite directions
#d)
#v is not a multiple of this vector - so, they are not parallel, and they do not point in the same direction

#Exercise 19:
eq:= -2*[8, 11, 3] + 4*[2, 1, 1];
#Result: eq := [-8, -18, -2]

#Exercise 25:
#u[1]/v[1] = 2
#u[2]/v[2] = -2
#u[3]/v[3] = -2
#u and v are not parallel, because u is not a multiple of v, and vice versa

#Exercise 27:
#u[1]/v[1] = -0.5
#u[2]/v[2] = -0.5
#u[3]/v[3] = 0.5
#u and v are not parallel, because u is not a multiple of v, and vice versa

#Exercise 31:
#We first find the unit vector of V
Magnitude := (V) -> sqrt(V[1]^2 + V[2]^2 + V[3]^2);
UV := [-4,4,2]/Magnitude([-4,4,2]);
#Result: UV := [-2/3, 2/3, 1/3]
#To reverse the direction, we multiply the unit vector by -1
UOpp:= -1*UV;
#Result: UOpp := [2/3, -2/3, -1/3]

#Exercise 49:
#Parametrization equation: R(t) = P + t*V
#Equation 1 -> R(t) = (5,5,2) + t*[0,-2,1]
#Because a vector parallel to v points in the same direction, we can use a multiple of v (2v, for example) for the second equation
#Equation 2 -> R(t) = (5,5,2) + t*[0,-4,2]

#Exercise 51:
R1:= [-1,2,2] + t*[4,-2,1];
R2:= [0,1,1] + t*[2,0,1];
#We expand both equations
expand(R1);
#Result: [4*t - 1, -2*t + 2, t + 2]
expand(R2);
#Result: [2*t, 1, t + 1]
#We come up with an equation in which these conditions are true:
eq:={R1[1]=R2[1], R1[2]=R2[2], R1[3]=R3[3]};
solve(eq,t);
#The result is empty - there is no solution, so, the lines never meet