To determine the relationship between the vectors [tex]\(u = -4i\)[/tex] and [tex]\(v = i\)[/tex], we need to consider their dot product and their direction. We will assess whether these vectors are orthogonal, parallel, or neither.
1. Dot Product:
The dot product of two complex vectors [tex]\(u\)[/tex] and [tex]\(v\)[/tex] is given by [tex]\(\langle u, v \rangle = \text{Re}(u \cdot \overline{v})\)[/tex].
Here, [tex]\(u = -4i\)[/tex] and [tex]\(v = i\)[/tex]. We need to compute the complex conjugate of [tex]\(v\)[/tex], which is [tex]\(\overline{v} = -i\)[/tex].
The dot product is then calculated as:
[tex]\[
\langle u, v \rangle = (-4i) \cdot (-i)
\][/tex]
We simplify this product:
[tex]\[
(-4i) \cdot (-i) = 4
\][/tex]
Thus, the real part of the dot product is:
[tex]\[
\text{Re}(4) = 4
\][/tex]
Since the dot product is non-zero (4), the vectors [tex]\(u\)[/tex] and [tex]\(v\)[/tex] are not orthogonal. Orthogonality requires the dot product to be zero.
2. Parallelism:
To determine if the vectors [tex]\(u\)[/tex] and [tex]\(v\)[/tex] are parallel, we need to check if one is a scalar multiple of the other.
We have [tex]\(u = -4i\)[/tex] and [tex]\(v = i\)[/tex].
We can write:
[tex]\[
u = (-4) \cdot v
\][/tex]
Since [tex]\(u\)[/tex] is indeed a scalar multiple of [tex]\(v\)[/tex] (with the scalar being [tex]\(-4\)[/tex]), the vectors [tex]\(u\)[/tex] and [tex]\(v\)[/tex] are parallel.
Therefore, the vectors [tex]\(u\)[/tex] and [tex]\(v\)[/tex] are parallel. The final answer is:
[tex]\[
\boxed{\text{Parallel}}
\][/tex]
