Final answer:
The vector sum vec a + vec b is a unit vector only if the angle between the two unit vectors vec a and vec b is 120 degrees (or 2pi/3 radians) since that is when cos(alpha) = -1/2.
Step-by-step explanation:
When determining whether the vector vec a + vec b is a unit vector for two given unit vectors vec a and vec b, and angle α between them, you can use the concept of vector addition and the dot product. The sum of two vectors can be expressed in terms of its magnitude, which is given by:
|vec a + vec b| = sqrt((vec a + vec b) · (vec a + vec b))
Since vec a and vec b are unit vectors, their magnitudes are 1 and the dot product vec a · vec b is given by cos(α). Hence:
|vec a + vec b|^2 = |vec a|^2 + |vec b|^2 + 2(vec a · vec b) = 1 + 1 + 2cos(α)
For vec a + vec b to be a unit vector as well, its magnitude must be 1. This leads to:
1 = 1 + 1 + 2cos(α)
Therefore, cos(α) must equal -1/2 for the sum of the vectors to have a magnitude of 1. In conclusion, vec a + vec b is a unit vector if the angle α between them is 120 degrees (or 2π/3 radians) since that is when cos(α) = -1/2.