Question

Let W be the subset of ℝ⁵ consisting of all vectors that are the same backwards as forwards. More precisely, a vector (v₁, v₂, v₃, v₄, v₅) in W satisfies:

Answer 1

The resultant vector when adding vectors u, V, and ŵ, each with a magnitude of 5 units and with u and ŵ being opposite, would have a magnitude of 5 units, as u and ŵ cancel each other out, leaving only vector V.

Step-by-step explanation:

When three vectors, u, V, and ŵ, each have the same magnitude and are oriented in specific directions to each other, the resultant vector's magnitude depends on their respective directions. Here, vector u points to the northwest, vector V points to the northeast, and vector ŵ points to the southwest, opposite to vector u. The addition of vectors is commutative which implies that the order of adding them does not affect the resulting vector.

Since vector ŵ is in the opposite direction to vector u, they cancel each other out. Thus, the resultant vector would have the same magnitude and direction as vector V alone. In this scenario, the magnitude of the resultant vector would be 5 units, since only two vectors cancel out, and the third retains its original magnitude and direction.