Answer:
k = (-55) / 8
k = (-3005) / 8
k = (-255 - sqrt(65025 - 510((-255 + sqrt(65025 - 510((-255 + sqrt(65025 - 510(0.309016^2))) / 2)^2)) / 2)^2)) / 2
k = (-255 - sqrt(65025 - 510((-255 + sqrt(65025 - 510((-255 + sqrt(65025 - 1469.59)))))^2)) / 2)
To find the acute angle between two vectors, we can use the dot product formula:
angle = arccos((a * b) / (||a|| * ||b||))
where a and b are the vectors, * is the dot product, and ||a|| and ||b|| are the magnitudes of the vectors a and b, respectively.
In this case, the dot product of a and b is (i - kj) * (i + j) = i^2 - kj * i + kj * i + kj^2 = 2i - k^2j
The magnitudes of the vectors a and b are ||a|| = sqrt(i^2 + (-kj)^2) = sqrt(1 + k^2) and ||b|| = sqrt(i^2 + j^2) = sqrt(2).
Substituting these values into the formula above, we get:
angle = arccos((2i - k^2j) / (sqrt(1 + k^2) * sqrt(2)))
Since the angle is given to be 60 degrees, we can set this equal to 60 degrees and solve for k:
60 = arccos((2i - k^2j) / (sqrt(1 + k^2) * sqrt(2)))
We can use the inverse cosine function to solve for k:
k = sqrt(1 / (cos(60)^2 - (2i / sqrt(1 + k^2) * sqrt(2))^2))
Since cos(60) = 0.5, we can substitute this value in and solve for k:
k = sqrt(1 / (0.5^2 - (2i / sqrt(1 + k^2) * sqrt(2))^2))
k = sqrt(1 / (0.25 - (2i / sqrt(1 + k^2) * sqrt(2))^2))
k = sqrt(1 / (0.25 - (4i^2 / (1 + k^2) * 2)^2))
k = sqrt(1 / (0.25 - (16 / (1 + k^2))^2))
k = sqrt(1 / (0.25 - 256 / (1 + k^2)^2))
k = sqrt((1 + k^2)^2 / (256 - (1 + k^2)^2))
k = sqrt((1 + k^4) / (256 - 1 - 2k^2 - k^4))
k = sqrt((k^4 + 1) / (255 - 2k^2))
We can then solve for the roots of this equation to find the possible values of k:
k = sqrt((k^4 + 1) / (255 - 2k^2))
k^4 - (255 - 2k^2)k^2 + 1 = 0
This is a quartic equation and can be solved using the quartic formula:
k = sqrt((-b +- sqrt(b^2 - 4ac)) / 2a)
where a, b, and c are the coefficients of the polynomial. In this case, a = 1, b = -(255 - 2k^2), and c = 1.
Substituting these values into the quartic formula, we get:
k = sqrt((-(-(255 - 2k^2)) +- sqrt((-(255 - 2k^2))^2 - 4 * 1 * 1)) / 2 * 1)
k = sqrt((255 - 2k^2 +- sqrt((255 - 2k^2)^2 - 4)) / 2)
k = sqrt((255 - 2k^2 +- sqrt(255^2 - 510k^2 + 4k^4)) / 2)
k = sqrt((255 - 2k^2 +- sqrt(255^2 - 510k^2)) / 2)
k = sqrt((255 - 2k^2 +- sqrt(65025 - 510k^2)) / 2)
Solving for the roots of this equation gives us the possible values of k:
k = (-255 + sqrt(65025 - 510k^2)) / 2
k = (-255 - sqrt(65025 - 510k^2)) / 2
The first equation gives us one possible value of k:
k = (-255 + sqrt(65025 - 510k^2)) / 2
Substituting k = (-255 + sqrt(65025 - 510k^2)) / 2 into the second equation gives us the second possible value of k:
k = (-255 - sqrt(65025 - 510((-255 + sqrt(65025 - 510k^2)) / 2)^2)) / 2
Simplifying this expression gives us the final possible value of k:
k = (-255 - sqrt(65025 - 510((-255 + sqrt(65025 - 510((-255 + sqrt(65025 - 510k^2)) / 2)^2)) / 2)^2)) / 2
Therefore, the possible values of k are:
k = (-255 + sqrt(65025 - 510k^2)) / 2
k = (-255 - sqrt(65025 - 510((-255 + sqrt(65025 - 510k^2)) / 2)^2)) / 2
solve for k in each
To solve for k in the first equation, we can isolate k by moving everything else to the right side of the equation:
k = (-255 + sqrt(65025 - 510k^2)) / 2
2k = -255 + sqrt(65025 - 510k^2)
2k + 255 = sqrt(65025 - 510k^2)
(2k + 255)^2 = 65025 - 510k^2
4k^2 + 1020k + 65025 = 65025 - 510k^2
4k^2 + 1530k + 65025 = 0
This is a quadratic equation, and we can use the quadratic formula to solve for k:
k = (-b +- sqrt(b^2 - 4ac)) / 2a
where a, b, and c are the coefficients of the polynomial. In this case, a = 4, b = 1530, and c = 65025.
Substituting these values into the quadratic formula gives us:
k = (-1530 +- sqrt(1530^2 - 4 * 4 * 65025)) / 2 * 4
k = (-1530 +- sqrt(3080400 - 2601000)) / 8
k = (-1530 +- sqrt(477900)) / 8
k = (-1530 +- sqrt(222725)) / 8
k = (-1530 + 1475) / 8
k = (-55) / 8
k = (-1530 - 1475) / 8
k = (-3005) / 8
Therefore, the solutions to the first equation are:
k = (-55) / 8
k = (-3005) / 8
