The vectors u, v are given by u = 3i + 5j, v = i - 2j. Find scalars a, b such that a(u + v) = 8i + (b - 2)j
To find scalars a and b, we need to solve the equation a(u + v) = 8i + (b - 2)j using the given vectors u and v.
First, let's expand the expression a(u + v) using the distributive property:
a(u + v) = au + av
Substituting the given values of u and v:
au + av = (3ai + 5aj) + (ai - 2aj)
Now, let's combine like terms:
au + av = (3a + a)i + (5a - 2a)j
Simplifying further:
au + av = (4a)i + (3a)j
From the right side of the equation 8i + (b - 2)j, we can deduce:
4a = 8 (equating the i components)
3a = b - 2 (equating the j components)
Solving the first equation for a:
4a = 8
a = 8/4
a = 2
Substituting the value of a in the second equation:
3(2) = b - 2
6 = b - 2
b = 6 + 2
b = 8
Therefore, the scalars a and b that satisfy the equation are a = 2 and b = 8.