Question Video: Identifying When Vector Addition Is Commutative Mathematics

When is it true that 𝐮 + 𝐯 = 𝐯 + 𝐮? [A] only when 𝐮 and 𝐯 are equivalent [B] only when 𝐮 and 𝐯 are parallel [C] only when 𝐮 and 𝐯 are perpendicular [D] only when 𝐮 and 𝐯 are not perpendicular [E] for any vectors 𝐮 and 𝐯

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Video Transcript

When is it true that vector 𝐮 plus vector 𝐯 is equal to vector 𝐯 plus vector 𝐮? Is it (A) only when vectors 𝐮 and 𝐯 are equivalent? (B) Only when vectors 𝐮 and 𝐯 are parallel. (C) Only when vectors 𝐮 and 𝐯 are perpendicular. (D) Only when vectors 𝐮 and 𝐯 are not perpendicular. Or (E) for any vectors 𝐮 and 𝐯.

Let’s begin by letting vector 𝐮 have components 𝑥 one, 𝑦 one and vector 𝐯 have components 𝑥 two, 𝑦 two. We know that when adding vectors, we simply add the corresponding components. Therefore, vector 𝐮 plus vector 𝐯 has components 𝑥 one plus 𝑥 two and 𝑦 one plus 𝑦 two. In the same way, vector 𝐯 plus vector 𝐮 has components 𝑥 two plus 𝑥 one and 𝑦 two plus 𝑦 one.

We know that the values of 𝑥 one, 𝑥 two, 𝑦 one, and 𝑦 two are all scalars and that adding two scalars is commutative. This means that 𝑥 one plus 𝑥 two will give us the same value as 𝑥 two plus 𝑥 one. Likewise, 𝑦 one plus 𝑦 two gives the same value as 𝑦 two plus 𝑦 one.

This means that the correct answer is option (E). Vector 𝐮 plus vector 𝐯 is equal to vector 𝐯 plus vector 𝐮 for any vectors 𝐮 and 𝐯. This leads us to a general rule that the addition of vectors is commutative.

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