Which of the following properties of vector operations is incorrect? Is it option (A) the vector 𝐚 plus the vector 𝐛 is equal to the vector 𝐛 plus the vector 𝐚? Is it option (B) the vector 𝐚 added to the sums of vectors 𝐛 and 𝐜 is equal to the sum of vectors 𝐚 and 𝐛 added to vector 𝐜? Is it option (C) the vector 𝐚 plus the zero vector is equal to vector 𝐚? Is it option (D) the scalar 𝑐 plus 𝑑 multiplied by the vector 𝐚 is equal to 𝑐 times 𝑑 multiplied by the vector 𝐚? Or is it option (E) the vector 𝐚 added to negative vector 𝐚 is equal to the zero vector?
In this question, we’re given five statements which are potential properties of the various vector operations. We need to determine which of the five given options is incorrect. And there’s several different ways of doing this. The easiest way is just to recall all of the given properties of the vector operations.
First, we can recall the vector addition is commutative. This means for any two vectors 𝐚 and 𝐛 of the same dimension, 𝐚 plus 𝐛 is equal to 𝐛 plus 𝐚. And this is a property given in option (A). Next, we can also recall the vector addition is associative. This means for any three vectors of the same dimensions, 𝐚, 𝐛, and 𝐜, we have 𝐚 plus the sum of 𝐛 and 𝐜 is equal to the sum of 𝐚 and 𝐛 added to the vector 𝐜. In other words, we can add the vectors in any order we want. And this is the property given in option (B).
Next, we can also recall that adding the zero vector to any vector of the same dimension won’t change its value. In other words, for any vector 𝐚 and zero vector of the same dimension, 𝐚 plus zero is equal to 𝐚. And we call the zero vector the additive identity of vector addition. Therefore, option (C) is also a correct property of vector operations.
Finally, we can also recall the additive inverse property of vector addition. This tells us for any vector 𝐚, 𝐚 added to negative 𝐚 will be equal to the zero vector. And this is enough to answer our question if we assume that one of the given options is incorrect.
However, for due diligence, let’s also show that this property does not hold true. We’ll do this with an example. Let’s set 𝑐 equal to one, 𝑑 equal to three, and 𝐚 equal to the vector one, zero. We can now evaluate each side of the equation separately. Let’s start with 𝑐 plus 𝑑 multiplied by 𝐚. That’s one plus three times the vector one, zero. One plus three is equal to four. And remember, to multiply a vector by a scalar, we multiply all of the components of the vector by the scalar. This gives us the vector four times one, zero. And four times one is four. So this is equal to the vector four, zero.
Let’s now do the same for the right-hand side of this equation. We get 𝑐 times 𝑑 multiplied by vector 𝐚 is one times three times the vector one, zero. This time, we have that one times three is equal to three. And now we need to multiply each of the components of our vector by the scalar value of three. This then gives us the vector three, zero. And we can then see this is not the same as the other vector.
Therefore, for these values of 𝑐, 𝑑, and the vector 𝐚, we were able to show that 𝑐 plus 𝑑 times the vector 𝐚 is not equal to 𝑐 times 𝑑 times vector 𝐚. In other words, only the property given in option (D) is not a correct property of the vector operations.