Question Video: Properties of Operations on Vectors Mathematics

Video Transcript

Given that vector 𝐚 equals one, five and vector 𝐛 equals six, two, find 𝐚 plus 𝐛 plus negative 𝐚.

We can answer this question directly using the properties of vector addition. Firstly, using the commutative property, which states that vector 𝐮 plus vector 𝐯 is equal to vector 𝐯 plus vector 𝐮, we can rewrite our expression 𝐚 plus 𝐛 plus negative 𝐚 as 𝐚 plus negative 𝐚 plus 𝐛. Next, we’ll use the additive inverse property, which states that vector 𝐮 plus negative vector 𝐮 is equal to the zero vector. Applying this to our expression, vector 𝐚 plus negative vector 𝐚 is equal to the zero vector. So we’re left with the zero vector plus vector 𝐛.

Finally, we’ll use the additive identity property, which states that vector 𝐮 plus the zero vector is equal to vector 𝐮. This means that, in our question, the zero vector plus vector 𝐛 is simply equal to vector 𝐛. We are told in the question that vector 𝐛 is equal to six, two. This means that 𝐚 plus 𝐛 plus negative 𝐚 is also equal to six, two. A second method here would be to simply work with the components of vector 𝐚 and vector 𝐛. We need to add the vectors one, five and six, two and then add the negative of the vector one, five. We can distribute the negative over the vector by multiplying all of its components by negative one. Therefore, the third vector becomes negative one, negative five.

We can now just add the three vectors by finding the sum of their corresponding components. We begin by adding one, six, and negative one. This is equal to six. We then add the 𝑦-components of five, two, and negative five, giving us two. This confirms the answer we got using the properties of vector addition. Vector 𝐚 plus vector 𝐛 plus negative vector 𝐚 is equal to six, two.