Vectors A and B are added to produce C, that is A+B=C. If B=-5x+5y and C=2x-3y. Find the unit vector of A, that is find A (in component form).
To find the unit vector A, we first need to solve the equation A + B = C for A.
Given B = -5x + 5y and C = 2x - 3y, we can rewrite the equation as:
A + (-5x + 5y) = 2x - 3y
Now let's separate the variables into their respective components:
A_x + (-5x) + A_y + 5y = 2x - 3y
Combining like terms:
(A_x - 5x) + (A_y + 5y) = 2x - 3y
Simplifying further:
(A_x - 5x) + (A_y + 5y) = (2x - 3y)
Now, we can match the corresponding components on both sides of the equation:
A_x - 5x = 2x
A_y + 5y = -3y
Simplifying each equation:
A_x = 7x
A_y = -8y
Now, we have the components of A, but we need to convert it into a unit vector. To do that, we need to calculate the magnitude of A and divide each component by the magnitude.
The magnitude of a vector A with components (A_x, A_y) is given by:
|A| = sqrt(A_x^2 + A_y^2)
|A| = sqrt((7x)^2 + (-8y)^2)
|A| = sqrt(49x^2 + 64y^2)
|A| = sqrt(49(x^2) + 64(y^2))
Now, to find the unit vector A (in component form), we divide each component of A by its magnitude:
 = (A_x/|A|, A_y/|A|)
 = (7x/sqrt(49(x^2) + 64(y^2)), -8y/sqrt(49(x^2) + 64(y^2)))
Therefore, the unit vector A is given by:
A = (7x/sqrt(49(x^2) + 64(y^2)), -8y/sqrt(49(x^2) + 64(y^2)))