Note: u,v,w,a,b are all vectors
Given that
u= xa + 2yb
v= -2ya + 3yb
w= 4a - 2b
where a and b are collinear, find the values of x and y for which 2u - v =w
To find the values of x and y for which 2u - v = w, we need to substitute the given expressions for u, v, and w into the equation and solve for x and y.
Let's start by substituting the expressions for u, v, and w:
2u - v = w
2(xa + 2yb) - (-2ya + 3yb) = 4a - 2b
Now distribute the coefficients to expand the expression:
2xa + 4yb + 2ya - 3yb = 4a - 2b
Combine the like terms:
(2x + 2y)a + (4y - 3y)b = 4a - 2b
Since a and b are collinear, they can be written as multiples of each other:
a = kb
Substitute this relationship into the equation:
(2x + 2y)kb + (4y - 3y)b = 4a - 2b
Simplify the equation:
(2x + 2y + (4y - 3y))kb + 2b = 4a - 2b
Combine the terms with b:
(2x + 2y + b(4y - 3y))k = 4a - 4b
Since this equation must hold true for any value of k, the coefficients on both sides must be equal. Let's equate the coefficients:
2x + 2y + b(4y - 3y) = 4
2 - 2b = -4
Solve the second equation for b:
2b = 6
b = 3
Substitute this value for b back into the first equation:
2x + 2y + 3(4y - 3y) = 4
2x + 2y + 3y = 4
2x + 5y = 4
At this point, we have one equation with two unknowns (x and y). Without additional information, we cannot determine the exact values of x and y. However, we can express one variable in terms of the other.
Let's solve the equation for x:
2x = 4 - 5y
x = (4 - 5y)/2
If you have any specific values for a or b, you can substitute them into the equation to find the corresponding values for x and y.