Essential Questions: 1)What are the features of the 3D coordinate system? 2) How can we solve problems involving distance and midpoint formulas in the 3D coordinate system? 6.4 (orange geometry book)

The Arrangement of the Axes

______ axis _______ axis Right Handed System INDEX FINGER: ___________ MIDDLE FINGER: __________ THUMB: ___________

EXAMPLE: Locate the point P(1, 2, 3) in a 3-D coordinate system. *How to graph in the 3-D plane* STEP 1:Starting at the origin, count 1 unit in the positive direction along the x-axis. Make a mark on the x-axis at this position. STEP 2: From your mark on the x-axis, count 2 units in the positive direction along the y-axis, drawing a dashed line to represent the distance. Make a mark on the new position. STEP 3: From the new position, count 3 units in the positive direction along the z-axis, drawing a dashed line to represent the distance. Label the final position at point P(1, 2, 3).

EXAMPLE: Locate the point P(2,3,6) in a 3-D coordinate system. *How to graph in the 3-D plane* STEP 1:Starting at the origin, count 1 unit in the positive direction along the x-axis. Make a mark on the x-axis at this position. STEP 2: From your mark on the x-axis, count 2 units in the positive direction along the y-axis, drawing a dashed line to represent the distance. Make a mark on the new position. STEP 3: From the new position, count 3 units in the positive direction along the z-axis, drawing a dashed line to represent the distance. Label the final position at point P(1, 2, 3).

The x and y axes divide the plane into 4 _________________, so the x, y and z-axes will divide space into ___________ ____________________. Octant #LocationCoordinates 1 1 st Octant ( ) 2 Top – front – left ( ) 3 Top – back – right ( ) 4 Top – back – left ( ) 5 Bottom – front – right ( ) 6 Bottom – front – left ( ) 7 Bottom – back – right ( ) 8 Bottom – back – left ( )

There are three coordinate planes, and each plane is named by the pair of axes that determines the plane: In the xy-plane, the ______________ of every point is zero. In the xz-plane, the ______________ of every point is zero. In the yz-plane, the ______________ of every point is zero.

The Distance Formula in 3-D The distance, d, between the points (x 1, y 1, z 1 ) and (x 2, y 2, z 2 ) is given by Example: Find the distance between points R(4, 6, -9) and S (-3, 2, -6). Example: Find the distance between points P(2, -4, -2) and Q(0, -3, 1).

The Midpoint Formula in 3-D The midpoint of a segment with endpoints at (x 1, y 1, z 1 ) and (x 2, y 2, z 2 ) is the point: Example: Find the midpoint of the segment with the given endpoints. a.(5, -2, 3) and (6, -7, 4) b. (3, 2, 1) and (1, 2, 3)