Explanation
Given that a,b,c,a+b,b+c, and a+b+c are unit vectors, their magnitudes are 1.
Recall that for any unit vector v, ∣v∣2=1 and ∣v∣2=v⋅v=1.
Step 1: Use the property of ∣a+b∣2
Given ∣a+b∣=1:
∣a+b∣2=(a+b)⋅(a+b)=1
∣a∣2+∣b∣2+2(a⋅b)=1
Since ∣a∣=1 and ∣b∣=1:
1+1+2(a⋅b)=1
2+2(a⋅b)=1
2(a⋅b)=−1
a⋅b=−21
Step 2: Relate the dot product to the angle
The dot product is defined as a⋅b=∣a∣∣b∣cosθ, where θ is the angle between vectors a and b.
−21=(1)(1)cosθ
cosθ=−21
Step 3: Determine the angle θ
Since cosθ=−21 and the cosine function is negative in the second quadrant:
θ=arccos(−21)=32π
Therefore, the correct option is (d).
