To determine whether the vectors [tex]\( u \)[/tex] and [tex]\( v \)[/tex] are parallel or orthogonal, we need to consider their dot product. The two vectors [tex]\( u \)[/tex] and [tex]\( v \)[/tex] are given as:
[tex]\[ u = (6, -9) \][/tex]
[tex]\[ v = (-24, 36) \][/tex]
The dot product [tex]\( u \cdot v \)[/tex] is calculated using the formula:
[tex]\[ u \cdot v = u_1 v_1 + u_2 v_2 \][/tex]
Thus, we substitute the components of the vectors [tex]\( u \)[/tex] and [tex]\( v \)[/tex]:
[tex]\[ u \cdot v = (6) \cdot (-24) + (-9) \cdot (36) \][/tex]
Carrying out the multiplication, we get:
[tex]\[ u \cdot v = 6 \cdot (-24) + (-9) \cdot 36 \][/tex]
[tex]\[ u \cdot v = -144 + (-324) \][/tex]
[tex]\[ u \cdot v = -144 - 324 \][/tex]
[tex]\[ u \cdot v = -468 \][/tex]
The result of the dot product is:
[tex]\[ u \cdot v = -468 \][/tex]
Since [tex]\( u \cdot v = -468 \neq 0 \)[/tex], the vectors [tex]\( u \)[/tex] and [tex]\( v \)[/tex] are not orthogonal. Vectors are orthogonal if and only if their dot product is zero.
To determine if the vectors are parallel, we check if one is a scalar multiple of the other. A vector [tex]\( u = (u_1, u_2) \)[/tex] is parallel to [tex]\( v = (v_1, v_2) \)[/tex] if there exists a scalar [tex]\( k \)[/tex] such that:
[tex]\[ (u_1, u_2) = k(v_1, v_2) \][/tex]
Checking this condition:
[tex]\[ 6 = k \cdot (-24) \][/tex]
[tex]\[ -9 = k \cdot 36 \][/tex]
Solving for [tex]\( k \)[/tex] using the first component:
[tex]\[ k = \frac{6}{-24} \][/tex]
[tex]\[ k = -\frac{1}{4} \][/tex]
We should verify this value of [tex]\( k \)[/tex] with the second component:
[tex]\[ -9 = \left(-\frac{1}{4}\right) \cdot 36 \][/tex]
[tex]\[ -9 = -9 \][/tex]
Since the same [tex]\( k \)[/tex] satisfies both components, the vectors [tex]\( u \)[/tex] and [tex]\( v \)[/tex] are parallel. Parallel vectors may have a cosine of the angle [tex]\( \theta \)[/tex] between them equal to -1, indicating that they are pointing in exactly opposite directions.
In conclusion:
The vectors are parallel because [tex]\( \cos \theta = -1 \)[/tex].
