If vector 𝐀 is five 𝐢 minus two 𝐣 minus 𝐤 and vector 𝐁 equals negative 𝐣 plus two 𝐤, determine the magnitude of 𝐀 plus 𝐁 and the magnitude of 𝐀 plus the magnitude of 𝐁.
When we looked at this vertical bar notation, we said magnitude. So let’s remind ourselves what it means to find the magnitude of a vector given in three dimensions. Suppose we have the vector 𝐀 given by 𝑥𝐢 plus 𝑦𝐣 plus 𝑧𝐤. The magnitude of this vector describes its length and is given by the square root of 𝑥 squared plus 𝑦 squared plus 𝑧 squared. So we’re going to need to work out a few different things. First, we’ll find the sum of 𝐀 and 𝐁 and find their magnitude. Then, separately, we’ll find the magnitude of 𝐀 and add it to the magnitude of 𝐁.
So we find the sum of 𝐀 and 𝐁. It’s five 𝐢 minus two 𝐣 minus 𝐤 plus negative 𝐣 plus two 𝐤. And of course to add vectors, we simply add their components. This gives us the sum of our vectors as being five 𝐢 minus three 𝐣 plus 𝐤. And so to find the magnitude of their sum, we find the square root of the sum of the squares of each component, so the square root of five squared plus negative three squared plus one squared. That of course is the square root of 25 plus nine plus one, which is the square root of 35.
Now that we’ve calculated the magnitude of the sum, let’s find the sum of their magnitudes. For vector 𝐀, that’s the square root of five squared plus negative two squared plus negative one squared, which gives us the square root of 30. Then, the magnitude of vector 𝐁 is the square root of negative one squared plus two squared, which is root five. This means we can write the magnitude of 𝐀 plus the magnitude of 𝐁 as root 30 plus root five, or in ascending order root five plus root 30. And this serves as a useful reminder that the magnitude of the sum of two vectors is not equal to the sum of their magnitudes. And so we find that the magnitude of 𝐀 plus 𝐁 is root 35, whilst the magnitude of 𝐀 plus the magnitude of 𝐁 is root five plus root 30.