If vector a is perpendicular to vector b, then their dot product is zero:
a · b = 0
We can write vector b as:
b = x i + 4 j + z k
where x and z are unknown coefficients.
To find b, we need to solve for x and z such that the dot product of a and b is zero. We have:
a · b = (3i - 2j) · (x i + 4 j + z k)
= 3x - 8 + 0
= 3x - 8
Since a · b = 0, we get:
3x - 8 = 0
Solving for x, we get:
x = 8/3
Therefore, vector b is:
b = (8/3) i + 4 j + z k
where z is any real number.