We are given the vectors:
AB⃗ = 2a⃗ + b⃗
AD⃗ = a⃗ - 2b⃗
These represent adjacent sides of a parallelogram. The diagonal BD⃗ can be found using the vector addition property of a parallelogram:
BD⃗ = AB⃗ - AD⃗
Substituting the given values:
BD⃗ = (2a⃗ + b⃗) - (a⃗ - 2b⃗)
= 2a⃗ + b⃗ - a⃗ + 2b⃗
= (2a⃗ - a⃗) + (b⃗ + 2b⃗)
= a⃗ + 3b⃗
Now, to find the magnitude |BD⃗|:
|BD⃗| = |a⃗ + 3b⃗|
Using the magnitude formula for two vectors:
|P⃗ + Q⃗| = √(|P⃗|² + |Q⃗|² + 2|P⃗||Q⃗| cosθ)
Substituting |a⃗| = 1, |b⃗| = 1, and θ = 60°:
|BD⃗| = √(1² + 3² + 2(1)(3)cos60°)
= √(1 + 9 + 6 × (1/2))
= √(1 + 9 + 3)
= √13
Thus, the length of the diagonal BD⃗ is √13.
