# algebra

Show that the four points (2,0,1), (−1,2,3), (3,2,2) and (3,−6,−3) lie in a plane.

1. 👍 0
2. 👎 0
3. 👁 241
1. we know that 3 points define a plane. So, pick 3 points and determine the normal direction to their plane.

Now replace any of the points with the 4th point, and determine the new normal.

If the two normals are the same, the 4 points are coplanar.

If we label the direction vectors of the points u,v,w,x, then we need

(w-u)x(w-v) to be parallel to (x-u)x(x-v)

<1,2,1>x<4,0,-1> = <-2,5,-8>
<1,-6,-4>x<4,-8,-6> = <4,-10,16>

the two normals are parallel, so the 4 points are coplanar.

1. 👍 0
2. 👎 0
2. let's take the first 3 points and find the equation of the plane containing them
direction vector #1: (-3,2,2)
direction vector #2: (1,2,1)

normal to these , take the cross-product
= (-2, 5, -8)
or (2, -5, 8) in the opposite direction

equation of plane is
2x - 5y + 8z = c
but (2,0,1) lies on it
4 - 0 + 8 = c = 12

the plane equation using the first three points is
2x - 5y + 8z = 12
test for the 4th point (3,-6,-3)

LS = 2(3) -5(-6) + 8(-3) = 12 = RS

So, yes, all 4 points are coplanar.

1. 👍 0
2. 👎 0

## Similar Questions

1. ### math

3.Graph the points A(–5, 0 ), B(–4, 3), and C(0, –4) on the same coordinate plane. 2. Without graphing, identify the quadrant in which the point (x, y) lies if x < 0 and y < 0. (1 point) 3.Determine which ordered pair is a

2. ### Math

Rewrite the following definition as a biconditional: Points that lie on the same line are collinear. A.) If a point lies on a line, then it is collinear. B.) If a point lies on a line with another point, then the two points are

3. ### Math - repost

The angles of elevation θ and ϕ to an airplane are being continuously monitored at two observation points A and B, respectively, which are 5 miles apart, and the airplane is east of both points in the same vertical plane.

4. ### Math

Points C(−5, 8) and D(2,5) lie on line CD. If points C′ and D′ are created by translating points C and D left 6 units, what is the slope of line C′D′? A. 3/7 B. −3/7 C. 7/3 D. −7/3

1. ### geometry

Decide which one of the following statements is false. a. any three points lie on a distinct line. b. three noncollinear points determine a plane. c. a line contains at least two points. d. through any two distinct points there

2. ### Maths

Show that the points (20°N,20°E) and (60°N,160°W) lie on the same great circle.Find their great circle distance apart.

3. ### Physics

An air traffic controller observes two airplanes approaching the airport. The displacement from the control tower to plane 1 is given by the vector A , which has a magnitude of 220 km and points in a direction 32 degrees north of

4. ### Geometry

Plane R contains points ,M,and Y. Plane Z intersects with plane R at SM, AY intersects plane R at point Y. . Point A is not in plane Z or plane R. MY forms a right angle with MS. A. What is measure of < SMY? B. Name three points

1. ### geometry

Which of the following statements best describe Euclid's axioms about lines? A. Two distinct lines intersect in an inifinite number of points. B. A line contains a finite number of points. C. Lines contain exactly two points. D.

2. ### Calculus

Consider the solid that lies above the square (in the xy-plane) R=[0,2]*[0,2], and below the elliptic paraboloid z=100−x^2−4y^2. (A) Estimate the volume by dividing R into 4 equal squares and choosing the sample points to lie

3. ### Math

The angles of elevation θ and ϕ to an airplane are being continuously monitored at two observation points A and B, respectively, which are 5 miles apart, and the airplane is east of both points in the same vertical plane.

4. ### Math

Use the distance formula to determine whether the points lie on the same line. (0,4),(7,-6),(-5,11). I know the distance formula, but I don't see how that will help me find out if they lie on the same line.