The x and y components of a vector A are 4 and -5 respectively and Vector B are -2 and 1 respectively. If A-B+3C=0, find the Cartesian coordinates for vector C?
(4-(-2)+3Cx)i + (-5-1+3Cy)j=0
check that, then solve for Cx
4+4=-3Cx or Cx=-8/3
-5-1=-3Cy
Cy=2
check all that, I did it in my head.
A=4x-5y
B=-2x+1y
--------
6x-6y
But A-B+3C=0
therefore, 3C= A-B
--- ---
3 3
C= A-B
---
3
But A-B = 6x -6y
therefore, A-B
the ---
3
6x-6y
-----
3
= 2x-2y
Therefore, C= 2×-2y
A=4x-5y
B=-2x+1y
--------
6x-6y
But A-B+3C=0
therefore,
C= A-B
---
3
But A-B = 6x -6y
therefore,
6x-6y
-----
3
= 2x-2y
Therefore, C= 2×-2y
To find the Cartesian coordinates of vector C, we need to solve the equation A - B + 3C = 0, where A, B, and C are vectors.
Given the x and y components of vectors A and B:
A = (4, -5) (x component = 4, y component = -5)
B = (-2, 1) (x component = -2, y component = 1)
Let's represent vector C as (x, y). Substituting all the values into the equation:
(4, -5) - (-2, 1) + 3(x, y) = 0
Now, we can simplify this equation component-wise:
(4 - (-2), -5 - 1) + (3x, 3y) = (0, 0)
(6, -6) + (3x, 3y) = (0, 0)
Now, we can separate the equation into two separate equations:
6 + 3x = 0 (equating the x components)
-6 + 3y = 0 (equating the y components)
Solving these equations, we get:
3x = -6
x = -6/3
x = -2
3y = 6
y = 6/3
y = 2
Therefore, the Cartesian coordinates of vector C are (-2, 2).