Vectors and the Geometry of Space 2015 Section 10.1 The Three Dimensional Coordinate System

In this lesson you will learn: 3 Space - The three-dimensional coordinate system Points in space, ordered triples The distance between two points in space The midpoint between two points in space The standard form for the equation of a sphere

Three-Dimensional Space Previously you studied vectors in the Cartesian plane or 2-dimensions, now we are going to expand our knowledge of vectors to 3-dimensions. Before we discuss vectors, let’s look at 3-dimensional space. To construct a 3-dimensional system, start with a yz plane flat on the paper (or screen). Next, the x-axis is perpendicular through the origin. (Think of the x-axis as coming out of the screen towards you.) For each axis drawn the arrow represents the positive end. z y x

This is considered a right-handed system. z This is considered a right-handed system. To recognize a right-handed system, imagine your right thumb pointing up the positive z-axis, your fingers curl from the positive x-axis to the positive y-axis. y x In a left-handed system, if your left thumb is pointing up the positive z-axis, your fingers will still curl from the positive x-axis to the positive y-axis. Below is an example of a left-handed system. z x Throughout this lesson, we will use right-handed systems. y

The 3-dimensional coordinate system is divided into eight octants The 3-dimensional coordinate system is divided into eight octants. Three planes shown below separate 3 space into the eight octants. The three planes are the yz plane which is perpendicular to the x-axis, the xy plane which is perpendicular to the z-axis and the xz plane which is perpendicular to the y-axis. Think about 4 octants sitting on top of the xy plane and the other 4 octants sitting below the xy plane. z y yz plane y xz plane z x z xy plane y x x

The 3-dimensional coordinate system is divided into eight octants as shown in the diagram.

Plotting Points in Space Notice we draw the x- and y-axes in the opposite direction X = directed distance from yz-plane to some point P Y= directed distance from xz-plane to some point P Z= directed distance from xy-plane to some point P (x,y,z) So, to plot points you go out or back, left or right, up or down

Plotting Points in Space Every position or point in 3-dimensional space is identified by an ordered triple, (x, y, z). Here is one example of plotting points in 3-dimensional space: z P (3, 4, 2) y The point is 3 units in front of the yz plane, 4 points in front of the xz plane and 2 units up from the xy plane. x

Here is another example of plotting points in space Here is another example of plotting points in space. In plotting the point Q (-3,4,-5) you will need to go back from the yz plane 3 units, out from the xz plane 4 units and down from the xy plane 5 units. z y Q (-3, 4, -5) x As you can see it is more difficult to visualize points in 3 dimensions.

Distance Between Two Points in Space The distance between two points in space is given by the formula: Take a look at the next two slides to see how we come up with this formula.

Consider finding the distance between the two points, . It is helpful to think of a rectangular solid with P in the bottom back corner and Q in the upper front corner with R below it at . Using two letters to represent the distance between the points, we know from the Pythagorean Theorem that PQ² = PR² + RQ² Q P R Using the Pythagorean Theorem again we can show that PR² = Note that RQ is .

Starting with PQ² = PR² + RQ² Make the substitutions: PR² = and RQ = Thus, PQ² = Or the distance from P to Q, PQ = Q P R That’s how we get the formula for the distance between any two points in space.

Find the distance between the points P(2, 3, 1) and Q(-3,4,2). We will look at example problems related to the three-dimensional coordinate system as we look at the different topics. Example 1: Find the distance between the points P(2, 3, 1) and Q(-3,4,2). Solution: Plugging into the distance formula:

Example 2: Find the lengths of the sides of triangle with vertices (0, 0, 0), (5, 4, 1) and (4, -2, 3). Then determine if the triangle is a right triangle, an isosceles triangle or neither. Solution: First find the length of each side of the triangle by finding the distance between each pair of vertices. (0, 0, 0) and (5, 4, 1) (0, 0, 0) and (4, -2, 3) (5, 4, 1) and (4, -2, 3) These are the lengths of the sides of the triangle. Since none of them are equal we know that it is not an isosceles triangle and since we know it is not a right triangle. Thus it is neither.

Isosceles Triangle You Try: Find the lengths of the sides of triangle with vertices (1, -3, -2), (5, -1, 2) and (-1, 1, 2). Then determine if the triangle is a right triangle, an isosceles triangle or neither. Isosceles Triangle

The Midpoint Between Two Points in Space The midpoint between two points, is given by: Each coordinate in the midpoint is simply the average of the coordinates in P and Q. Example 3: Find the midpoint of the points P(2, 3, 0) and Q(-4,4,2).

You Try: Find the midpoint of the points P(5, -2, 3) and Q(0,4,4).

Equation of a Sphere A sphere is the collection of all points equal distance from a center point. To come up with the equation of a sphere, keep in mind that the distance from any point (x, y, z) on the sphere to the center of the sphere, is the constant r which is the radius of the sphere. Using the two points (x, y, z), and r, the radius in the distance formula, we get: If we square both sides of this equation we get: The standard equation of a sphere is where r is the radius and is the center. Points satisfying the equation of a sphere are “surface points”, not “interior points.”

Example 4: Find the equation of the sphere with radius, r = 5 and center, (2, -3, 1). Solution: Just plugging into the standard equation of a sphere we get: Example 5: Find the equation of the sphere with endpoints of a diameter (4, 3, 1) and (-2, 5, 7). Solution: Using the midpoint formula we can find the center and using the distance formula we can find the radius. Thus the equation is:

You Try: Find the equation of the sphere with endpoints of a diameter (2, -2, 2) and (-1, 4, 6).

Example 6: Find the center and radius of the sphere, . Solution: To find the center and the radius we simply need to write the equation of the sphere in standard form, . Then we can easily identify the center, and the radius, r. To do this we will need to complete the square on each variable. Thus the center is (2, -3, -4) and the radius is 6.

Find the center and radius of the sphere, You Try Find the center and radius of the sphere, Thus the center is (1/2, 4, -1) and the radius is 3

Traces The intersection of a sphere (or anything) with one of the three coordinate planes is called a trace. A trace occurs when the sphere is sliced by one of the coordinate planes. The trace of a sphere is a circle. To find the xy-trace, use the fact that every point in the xy-plane has a z-coordinate of 0. Substitute z=0 into the original equation and the resulting equation will represent the intersection of the sphere with the xy-plane.

Example 7: Find the xy-trace of the sphere given by:

Example 8: You Try: Find the yz-trace of the sphere given by:

Homework Day 1: Pg.711 1-9, 17-55 odds Day 2: Pg.711 2-10, 18-56 evens