3 Dimensional Space Name: Graph paper is only so helpful with 3 D graphing, because it doesn't give a grid in the z dimension like the first image below is attempting to do. The image on the right is showing how to find a point in 3 D space. Of course you need three coordinates. (x,y,z) a) How many different pathways are there from the origin to the point P in the picture above? Keep in mind that the movements need to follow an axis. So you can't go diagonally through space (moving on two dimensions at once). Each movement must be in one dimension (either x or y or z, then the next, and then the next). Notice the picture on the right is showing one possible pathway...x 1st, then z 2nd, and y 3rd. Keep in mind that you can do this path in different orders. Draw all the possible pathways as you count them, and you will draw a box. (Unless you are a good artist, pencil, straight edge, and eraser is recommended.) The number of pathways:____________ b) In the first picture above try to graph the point (3,4,2) and label it W. Draw the box as you did in part a to help with the perspective. c) Vectors exist in 3 D space too. Draw a line from the origin out to point W. Notice that this vector shoots through the box to the opposite corner. This would be the vector <3,4,2> which does not necessarily need to start from the origin. It can be moved anywhere within the 3D space. When it sits on the origin like this, it is referred to as the \"position vector.\" d) Considering the vector <3,4,2>, what would be the coordinates of the point at the end of two of these vectors put together (if the first one is the position vector)? Coordinates:_____________ 1. a) Draw the following 3D position vector <6,4,3>. This is the vector from the origin to the point (6,4,3). Include the pathway box drawn in lightly to help with the perspective. b) Finding distances in 3D space is similar to finding distances in 2D space. Use the picture above to help you find the magnitude (distance) of the vector. It will take a few steps with right triangles. Draw the triangles above as you show your algebraic work below: 2. a) Use the space below to draw a generic 3D vector . b) Now similar to the work from #1, derive the formula for the magnitude of any 3D vector 3. Consider what you have learned with vectors in 2D. In particular recall adding two vectors together tip to tail. We will try adding together two vectors in 3D space, vector A<4,3,2> and B<2,3,-3> a) Start by drawing vector A below b) At the tip of vector A place the tail of vector B. Draw vector B above. c) What is the coordinates of the point at the end of the two vectors? c) Now there is a new vector C which is the resultant of A+B. This is the vector from the beginning of A (at the origin) to the end of vector B. Draw it above. d) Give vector C: e) Drawing 3D vectors can be challenging, but adding them doesn't require you to be able to draw them. Describe in the space below how to add 3D vectors: f) Given two vectors M<1,-3,5> and N<6,-1,-7>, what new vector do you get from 2M + N