If vector 𝐀 is equal to 𝐢 plus 𝐣 plus 𝐤, vector 𝐁 is equal to 𝐢 minus two 𝐣 plus three 𝐤, and vector 𝐂 equals negative 𝐢 minus 𝐣 plus 𝐤, find 𝐀 plus 𝐁 minus 𝐂.
In order to add or subtract vectors written in terms of their 𝐢, 𝐣, and 𝐤 unit vectors, we simply add or subtract the corresponding components. We can begin with the 𝐢-components. We have 𝐢 plus 𝐢 minus negative 𝐢. 𝐢 plus 𝐢 is equal to two 𝐢. And subtracting negative 𝐢 from this gives us three 𝐢.
For the 𝐣-components, we need to add 𝐣 and negative two 𝐣 and then subtract negative 𝐣. 𝐣 plus negative two 𝐣 is equal to negative 𝐣. Subtracting negative 𝐣 from this gives us zero.
Finally, we have the 𝐤-components. We need to add 𝐤 and three 𝐤 and then subtract 𝐤. This is equal to three 𝐤. Vector 𝐀 plus vector 𝐁 minus vector 𝐂 is equal to three 𝐢 plus zero 𝐣 plus three 𝐤. This can be simplified to three 𝐢 plus three 𝐤.
If vector 𝐀 is equal to 𝐢 plus 𝐣 plus 𝐤, vector 𝐁 is equal to 𝐢 minus two 𝐣 plus three 𝐤, and vector 𝐂 is equal to negative 𝐢 minus 𝐣 plus 𝐤, then 𝐀 plus 𝐁 minus 𝐂 is equal to three 𝐢 plus three 𝐤.