(a) Find the initial point of the vector that is equivalent to u = (1, 2) and whose terminal point is B(2, 0).
(b) Find the terminal point of the vector that is equivalent to u = (1, 1, 3) and whose initial point is A(0, 2, 0).
Part (a): Find the initial point of the vector equivalent to u = (1, 2) with terminal point B(2, 0)
- Let’s denote the initial point of the equivalent vector v as A(x, y).
- The terminal point is given as B (2, 0).
- The vector v from A to B can be expressed as B – A = (2 – x , 0 – y).
- Since v is equivalent to u, their components must be equal:
- (2 − x , 0 − y ) = ( 1 , 2 )
- Solving for x and y:
- For the x-component:
- 2 – x = 1 ⟹ x = 1
- For the y-component:
- 0 – y = 2 ⟹ y = −2
- Conclusion: The initial point A is at (1, -2).
- Let’s verify by calculating the vector from A(1, -2) to B(2, 0):
- (2−1,0−(−2))=(1,2)
- This matches the vector u, confirming our solution is correct.
Part (b): Find the terminal point of the vector equivalent to u = (1, 1, 3) with initial point A(0, 2, 0)
- Let’s denote the terminal point of the equivalent vector v as B(x, y, z).
- The initial point is given as A(0, 2, 0).
- The vector v from A to B can be expressed as B – A = (x – 0, y – 2, z – 0).
- Since v is equivalent to u, their components must be equal:
- ( x , y−2 , z ) = ( 1 , 1 , 3 )
- Solving for x, y, and z:
- For the x-component:
- x =1
- For the y-component:
- y − 2 = 1 ⟹ y = 3
- For the z-component:
- z = 3
- The terminal point B is at (1, 3, 3).
- Verification: Let’s verify by calculating the vector from A(0, 2, 0) to B(1, 3, 3):
- (1−0 , 3−2 , 3−0 ) = ( 1 , 1 , 3 )
- This matches the vector u, confirming our solution is correct.