Answer: To find the values of "c" for which |v + w| = 3, we need to first calculate v + w and then find the magnitude (or length) of the resulting vector.
Given:
v = 5i - 5j
w = xi + 6j
Let's find v + w:
v + w = (5i - 5j) + (xi + 6j)
Now, combine like terms:
v + w = (5 + x)i + (6 - 5)j
v + w = (5 + x)i + j
Now, to find the magnitude of v + w, we use the formula:
|v + w| = √((5 + x)^2 + 1^2)
We want |v + w| to be equal to 3:
√((5 + x)^2 + 1^2) = 3
Now, let's solve for "x":
(5 + x)^2 + 1 = 3^2
(5 + x)^2 + 1 = 9
(5 + x)^2 = 9 - 1
(5 + x)^2 = 8
Now, take the square root of both sides:
5 + x = ±√8
5 + x = ±2√2
Now, isolate "x" in each case:
5 + x = 2√2
x = 2√2 - 5
5 + x = -2√2
x = -2√2 - 5
So, there are two values of "c" for which |v + w| = 3:
c = 2√2 - 5
c = -2√2 - 5