Part A:
To find the linear form of a vector, we need to subtract the initial point from the terminal point to get the components of the vector.
Vector u:
Initial point S(14,23) and terminal point T (5,19)
u = T - S
u = (5 - 14, 19 - 23)
u = (-9, -4)
Linear form: u = -9i - 4j
Vector v:
Initial point A (7,17) and terminal point B(32,9)
v = B - A
v = (32 - 7, 9 - 17)
v = (25, -8)
Linear form: v = 25i - 8j
Part B:
4u-5v = 4(-9i - 4j) - 5(25i - 8j)
= (-36i - 16j) - (125i - 40j)
= -161i + 24j
Part C:
Given vector t = -16i + 36j
To determine if t and u are parallel, orthogonal, or neither, we need to find the dot product of t and u:
t · u = (-16)(-9) + (36)(-4)
t · u = 144
Since the dot product t · u is not equal to 0, t and u are not orthogonal. To determine if they are parallel, we need to find the magnitude of t and u:
|t| = sqrt((-16)^2 + (36)^2) = 40
|u| = sqrt((-9)^2 + (-4)^2) = sqrt(97)
The directions of the two vectors are not the same, so they are not parallel. Therefore, t and u are neither parallel nor orthogonal.
Part D:
To find another vector w that has the same relationship to vector t as vector u, we need to find a vector w that is parallel to u and has the same magnitude as t.
We know that vector u is not parallel to t, so we need to find the projection of t onto u to get a vector that is parallel to u and has the same direction as t. The projection of t onto u is given by:
proj_u t = (t · u / |u|^2) u
Substituting the values we have:
proj_u t = (t · u / |u|^2) u
= (144 / 97) (-9i - 4j)
= (-16.727)i - (7.818)j
This vector is parallel to u and has the same direction as t. To find a vector with the same magnitude as t, we just need to scale this vector:
w = (|t| / |proj_u t|) proj_u t
w = (40 / |proj_u t|) (-16.727i - 7.818j)
w = (-22.688i - 10.559j)
Therefore, vector w = -22.688i - 10.559j has the same relationship to vector t as vector u, in the sense that they have the same magnitude and are parallel.