let u10 and v be a unit vector in the plane what are the pos

let u=[1,0] and v be a unit vector in the plane. what are the possible values of ||u+v||? give a unit vector v such that ||u+v|=3

Solution

Let V = xi + yj and given that U = i
given that V is unit vector so sqrt (x² + y² )=1
U+V = (x+1) i + y j
|| U+V| = Sqrt ( (x+1)² + y² )
= sqrt ( x² + 2x + 1 + y² )
= sqrt ( 2+2x)

Thus, the possible values depends on the \'x\' value in sqrt(2+2x)
If x=-1 then sqrt(2+2x) = sqrt(2+2(-1)) = sqrt (0) = 0
if x=1 then sqrt (2+2x) = sqrt (2+2*1) = sqrt(4) = 2
Thus, the possible values of ||U+V|| = [0,2] which means it can take any value between (and including) 0 to 2;

We are to find a unit vector \'V\' such that || U+V || = sqrt(3)
sqrt(2+2x) = sqrt(3)
2+2x = 3
2x= 3-2
2x=1 so x=1/2
If x=1/2 then x² + y² = 1²
y² = 1- (1/2)² = 1 - 1/4 =3/4
y= sqrt(3/4) = sqrt(3) /2

Thus, V would be i /2 + sqrt(3) j / 2
or V= [1/2 , sqrt(3) / 2]