(a) Find the terminal point of the vector that is equivalent to u = (1, 2) and whose initial point is A(1, 1).
(b) Find the initial point of the vector that is equivalent to u = (1, 1, 3) and whose terminal point is B(−1, −1, 2).
(a) Terminal point of the equivalent vector with initial point A(1, 1)
Given:
- Vector u = (1, 2)
- Initial point A = (1, 1)
- A vector from A(1, 1) to B(x, y) is given by:
AB =(x−1,y−1)
- Since AB is equivalent to u, we set:
(x−1, y−1) = (1 , 2)
- Equate the components:
x−1 = 1 ⟹ x = 2
y−1 = 2 ⟹ y = 3
- Therefore, the terminal point is: (2 , 3)
(b) Initial point of the equivalent vector with terminal point B(−1, −1, 2)
Given:
- Vector u = (1, 1, 3)
- Terminal point B = (−1, −1, 2)
A vector from A (x, y, z) to B(−1, −1, 2) is given by:
AB = (−1−x,−1−y,2−z)
Since AB is equivalent to u, we set:
( −1− x , −1− y , 2 − z ) = ( 1 , 1 , 3 )
- Equate the components:
-1 – x = 1 ⟹ x = − 2
− 1 – y = 1 ⟹ y = − 2
2 – z = 3 ⟹ z =–1
Therefore, the terminal point is: ( -2 , -2 , -1 )