U = (a,b)
V = (c,d)
where a,b,c,d are scalars and are in the set of real numbers.
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Using those definitions, we can say
U - 3V = 1*U + (-3)*V
U - 3V = 1*(a,b) + (-3)*(c,d)
U - 3V = (a,b) + (-3c,-3d)
U - 3V = (a-3c,b-3d)
Since U-3V also equals (1,5), which is given, this means
(a-3c,b-3d) = (1,5)
further breaking down to
a-3c = 1
b-3d = 5
Let's refer to these equations as (1) and (2).
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Similarly,
-U + V = -1*U + 1*V
-U + V = -1*(a,b) + 1*(c,d)
-U + V = (-a,-b) + (c,d)
-U + V = (-a+c,-b+d)
Since -U + V also equals (-5,3), this means
(-a+c,-b+d) = (-5,3)
further breaking down into
-a+c = -5
-b+d = 3
Let's refer to these equations as (3) and (4).
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Group equation (1) and (3) together to get this system
a-3c = 1
-a+c = -5
Adding those equations together leads to -2c = -4 so c = 2
If c = 2, then...
a-3c = 1
a-3(2) = 1
a-6 = 1
a-6+6 = 1+6
a = 7
So far we know that a = 7 and c = 2
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Now group equation (2) and (4) together and solve the system
b-3d = 5
-b+d = 3
Adding those equations leads to -2d = 8 so d = -4
If d = -4, then,
-b+d = 3
-b+(-4) = 3
-b-4 = 3
-b-4+4 = 3+4
-b = 7
b = -7
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To wrap up everything so far, we found that
a = 7
b = -7
c = 2
d = -4
making
U = (a,b) and V = (c,d)
turn into
U = (7,-7) and V = (2,-4)
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We have the coordinates of U and V, so we can now compute the angle theta between them.
First we'll need the dot product
U dot V = a*c + b*d
U dot V = 7*2 + (-7)*(-4)
U dot V = 14 + 28
U dot V = 42
Now we need the length of each vector
Let's find the length of vector U
|U| = sqrt(U dot U)
|U| = sqrt(a*a + b*b)
|U| = sqrt(a^2 + b^2)
|U| = sqrt(7^2 + (-7)^2)
|U| = sqrt(49 + 49)
|U| = sqrt(49*2)
|U| = sqrt(49)*sqrt(2)
|U| = 7*sqrt(2)
And we also need the length of vector V
|V| = sqrt(V dot V)
|V| = sqrt(c*c + d*d)
|V| = sqrt(c^2 + d^2)
|V| = sqrt(2^2 + (-4)^2)
|V| = sqrt(4 + 16)
|V| = sqrt(20)
|V| = sqrt(4*5)
|V| = sqrt(4)*sqrt(5)
|V| = 2*sqrt(5)
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It's a lot of work, but we're at the home stretch. Plug the dot product result, and the vector lengths, into the formula below to find
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Final Answer: 18.4349488229 degrees
The answer is approximate. Round it however you need to.
Side Note: The answer has been confirmed with GeoGebra (free graphing software) as shown below
The vectors "check1" and "check2" are defined to be
check1 = U - 3V
check2 = -U + V
so that part is confirmed as well.