### Lesson 10-5: Planes In Space

A, B, and C: coefficients of the components of a normal vectorAx + By + Cz =D

D: value determined after substituting the coordinates of a given point

First, let’s review the equation using a problem…

The figure below shows a plane in space with a vector , and a point P (4, 6, 8) both contained on the plane. How do we find the equation of the plane?

**NOTE:** To determine this equation, we have to find an equation of the plane relating x, y, and z where the vector is perpendicular to all points

**Step One:** Using the given vector equation and point, find a new point contained on the plane.

vector:

point a:

(4, 6, 8)

new point b:

(x, y, z)

**Step Two:** Now that we have two points, we can create a line connecting the point given to us (4,6,8) to point (x,y,z). As we learned from the mini-quiz, multiplying these two points together gives us a dot product of 0, which is equal to the cos(90°).

**Step Three:** Next we need to distribute the equation of our vector:

to the equation relating our two points (v).

**Step Four:**Then, we multiply both sets of equations and set the result equal to zero.

**Step Five:**With this last equation, we discovered that the coefficients were the same as those of the perpendicular vector equation:

Thus, to find the equation of the plane more efficiently, we can use the vector coefficients and replace them with A, B, and C. To find D, we simply distribute the vector coefficients to the corresponding point.

We distribute the coefficients 12, 3, and 14 to the corresponding x, y, and z values of the ordered pair (4, 6, 8). Thus, 12 x 4, 3 x 6, and 14 x 8. After adding up these numbers, we get 178. Thus, our final equation is:

178 = 12x + 3y + 14z

**Helpful Notes:**

- If you are given the equation for a plane and need to find the equation for the perpendicular vector, remember the coefficients are the same! All you have to do is replace x, y, z, with i, j, k !
- If you are given the equation for a plane and need to find a coordinate for your point on the plane, just plug in the coordinates for x, y, and z, and use basic algebra to solve!

**For More Info…**Although this website seems a little more complicated, the animation helps me visualize exactly what the equation and subsequent steps are for solving:

http://college.hmco.com/mathematics/larson/precalculus_limits/1e2/ins_resources/ap.html

**Fun Stuff…**

This is from my favorite comic site, toothpastefordinner.com. (its my favorite comic for the month):

http://www.toothpastefordinner.com/012207/latest-poll-results-what.gif

**Annie, you are up next! Good luck!**

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