Given that w = 8i + 3j, x = 6i - 5j, y = 2i + 3j and |z| = 41. find z in the direction of w + x - 2y.
To find the vector "z" in the direction of the sum of vectors "w", "x", and "-2y", we can first find the sum of "w", "x", and "-2y", then find the unit vector in that direction and multiply it by the magnitude of "z".
First, let's find the sum of "w", "x", and "-2y":
w = 8i + 3j
x = 6i - 5j
y = 2i + 3j
-2y = -4i - 6j
w + x - 2y = (8i + 3j) + (6i - 5j) - (-4i - 6j) = 18i - 8j
Next, let's find the unit vector in the direction of the sum:
The unit vector in the direction of a vector can be found by dividing the vector by its magnitude.
The magnitude of the sum is:
√(18^2 + (-8)^2) = 20
So the unit vector in the direction of the sum is:
(18i - 8j) / 20 = 9/10i - 4/10j
Finally, let's multiply the unit vector by the magnitude of "z" to find "z":
z = 41 * (9/10i - 4/10j) = 36.9i - 16.4j
So the vector "z" in the direction of the sum of "w", "x", and "-2y" is approximately 36.9i - 16.4j.