##### Like what you saw?

##### Create FREE Account and:

- Watch all FREE content in 21 subjects(388 videos for 23 hours)
- FREE advice on how to get better grades at school from an expert
- Attend and watch FREE live webinar on useful topics

# Vector Operations in 3D - Concept

###### Norm Prokup

###### Norm Prokup

**Cornell University**

PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

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.

Please enter your name.

Are you sure you want to delete this comment?

###### Norm Prokup

PhD. in Mathematics, University of Rhode Island

B.S. in Mechanical Engineering, Cornell University

He uses really creative examples for explaining tough concepts and illustrates them perfectly on the whiteboard. It's impossible to get lost during his lessons.

Thiswas EXCELLENT! I am a math teacher and have been looking for an easy/logical way to explain the lateral area of a cone to my students and this was incredibly helpful, thank you very much!”

I just learned more In 3 minutes of polygons here than I do in 3 weeks in my math class”

Hahaha, his examples are the same problems of my math HW!”

##### Concept (1)

##### Sample Problems (2)

Need help with a problem?

Watch expert teachers solve similar problems.

## Comments (0)

Please Sign in or Sign up to add your comment.

## ·

Delete