Suppose u=< 2,−1> and v=< 1,−8> are two vectors that form the sides of a parallelogram. then the lengths of the two diagonals of the parallelogram are?

| u | = √(2² + (-1²)) = √5
| v | = √ ( 1² + (-8)² = √65
cos (u,v) = ( u * v ) / (| u | * | v |) =
 (2 * 1 + ( -1 ) * ( - 8 )) / √5 √ 65 = (2 + 8) / √5 √65 = 10 / (√5 √ 65 )
The length of a larger diagonal:
d 1² = | u |² + 2 |u| |v| + | v |² = 5 + (2 √5 √65 * 10 / √5 √65 )+65
d 1² = 70 + 20 = 90
d 1 = √ 90 = 3√10
d 2² = 70 - 20 = 50
d 2 = √50 = 5√2

The lengths of the diagonals are: 3√10 and 5√2 .