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.