If Vector a = (2, 2k), Vector b = (4, -k), are perpendicular, solve for k
with full solutions?
To determine if two vectors are perpendicular, we use the dot product. For two vectors, a = (a1, a2) and b = (b1, b2), the dot product is given by a · b = a1 * b1 + a2 * b2.
In this case, the vectors a = (2, 2k) and b = (4, -k). Therefore, the dot product of a and b is:
a · b = (2)(4) + (2k)(-k)
= 8 - 2k^2
Since the vectors are perpendicular, their dot product is 0. Therefore, we have the equation:
8 - 2k^2 = 0
To solve for k, we can rearrange the equation as follows:
2k^2 = 8
k^2 = 4
k = ±√4
Hence, k can have two solutions: k = 2 and k = -2.
Therefore, the value of k that satisfies the condition for the vectors a and b to be perpendicular is k = 2 and k = -2.