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Vector Operations in 3D - Concept
Although they are similar to 2D vector operations, it is good to get practice doing 3D vector operations. 3D vector operations include addition and scalar multiplication, the dot product and the calculation of magnitude. The biggest difference in these 3D vector operations is an added step of computation. With 3D vector operations we can do computation such as find the angle between vectors in space.
So how do vector operations work in 3 dimensions? It turns out it's almost exactly the same way as two dimensions only you have a new component, now let's take 2 vectors u and v. In 3 dimensions you have 3 components u1, u2 and u3, v1, v2 and v3 so how do you add vectors? Just the same as in 2 dimensions you add them component wise so the sum will be u1+v1, u2+v2, u3+v3. What about scalar multiples? Suppose k is some real number and you're multiplying it by vector u, well same as before it's k times u1, k times u2, k times u3, k times each of the components. The dot product works exactly the same way too it's going to be the product of like components so u1v1+u2v2+u3v3, u1v1+u2v2+u3v3 and finally the magnitude of a vector u is just the square root of the sum of the squares of the components. So u1 squared plus u2 squared plus u3 squared okay.
Let's do an example that exercises these rules, so part a asks me to simplify 3, 8 negative 2 plus 2 times 4 negative 1 and 2. So I'm going to do the scalar multiplication first 2 times 4 negative 1, 2 is 8 negative 2, 4 and I'll add that to 3, 8 negative 2 so I get 3+8 is 11, 8 plus negative 2 is 6 and negative 2 plus 4 is 2. How about the dot product of these 2 guys it's going to be 1 times 3, 3 plus negative 4 times 2 negative 8 plus 5 times 6 30. And that's negative 5 plus 30 25, how does subtraction work? Again exactly the same as with 2 dimensions you subtract like components 1-2 negative 1, 2 minus negative 1 is 3, 1-1 0. What's the magnitude of the vector 3, 2, 6 it's the square root of the sum of the squares of the components. So 9+4+36, now this is 13+36 which is 49 and root 49 is 7.