Final answer:
The vectors u = 5<sub>j</sub> and v = -2<sub>j</sub> are orthogonal.
Step-by-step explanation:
To determine whether vectors u and v are orthogonal, we use the dot product. Two vectors are orthogonal if their dot product equals zero. For vectors u = 5<sub>j</sub> and v = -2<sub>j</sub>, where j represents a unit vector in the y-direction, the dot product u ⋅ v can be calculated as 5<sub>j</sub> ⋅ (-2)<sub>j</sub>.
Let's compute the dot product:
u ⋅ v = 5<sub>j</sub> ⋅ (-2)<sub>j</sub>
= 5 × (-2) × j ⋅ j
= -10 × (j ⋅ j)
Since j ⋅ j represents the dot product of a vector with itself, which is equivalent to the square of its magnitude, j ⋅ j = |j|² = 1. Therefore, u ⋅ v = -10 × (1) = -10. As the dot product u ⋅ v is not equal to zero, the vectors are not orthogonal. They are orthogonal if and only if their dot product is zero.
For vectors to be parallel, one must be a scalar multiple of the other. In this case, the vectors u and v have different magnitudes and opposite directions, indicating they are not parallel. Hence, the final determination is that the vectors u = 5<sub>j</sub> and v = -2<sub>j</sub> are orthogonal.