Vectors "v" and "w" are given by v = 5i - 2j and w = i + 3j. Find scalars r and s such that r ( v - w) = ( r + s )i - 20j.
r(v-w)=(r+s)i-20j
start with the i componnets.
r(v-w) dot i=r+s
(5ri-2rj-ri-3rj)dot i= r+s
5r-r=r+s
4r=s
Now the j components
r(v-w)dot j=-20
-2r-3r=-20
r=4 then s=1
check my work, please
WAT DO U MEAN BY DOT?
The answer given above is incorrrect
let v = [5,-2) and w = [1,3]
then v-w = [4,-5]
so we want r[4,-5] = [r+s, -20]
then 4r = r+s and -5r = -20
3r = s and r = 4
then r = 4, then s = 12
verification:
LS = r(v-w)
= 4[4,-5] = [16, -20]
RS = [r+s, -20 = 16, -20 ] = LS
My answer is correct!
To find the scalars r and s, we need to simplify the expressions on both sides of the equation and then compare the coefficients of i and j.
Let's start by simplifying r(v - w):
r(v - w) = r(5i - 2j - (i + 3j))
= r(5i - 2j - i - 3j)
= r(4i - 5j)
Now, let's simplify the right side of the equation:
(r + s)i - 20j
Since we want the coefficients of i and j on both sides to be equal, we need to set the coefficients of i and j in both expressions equal to each other.
Comparing the coefficients of i:
4r = r + s
Comparing the coefficients of j:
-5r = -20
Solving the equation -5r = -20 gives us r = 4.
Substituting r = 4 into the equation 4r = r + s gives us 4(4) = 4 + s, which simplifies to 16 = 4 + s.
Solving for s, we subtract 4 from both sides, giving us s = 16 - 4 = 12.
Therefore, the scalars r and s that satisfy the equation r(v - w) = (r + s)i - 20j are r = 4 and s = 12.