Two objects are coplanar if they both lie in the same plane. In the applet above, there are 16 coplanar points. They are coplanar because they all lie in the same plane as indicated by the yellow area. If you uncheck the 'coplanar' checkbox, the points are then randomly spread out in space and are therefore not coplanar*. In geometry, a set of points in space are coplanar if there exists a geometric plane that contains them all. For example, three points are always coplanar, and if the points are distinct and non-collinear, the plane they determine is unique. However, a set of four or more.
Definition. Vectors parallel to the same plane, or lie on the same plane are called coplanar vectors (Fig. 1).
Fig. 1 |
It is always possible to find a plane parallel to the two random vectors, in that any two vectors are always coplanar.
Condition of vectors coplanarity
- For 3-vectors. The three vectors are coplanar if their scalar triple product is zero.
- For 3-vectors. The three vectors are coplanar if they are linearly dependent.
- For n-vectors.Vectors are coplanar if among them no more than two linearly independent vectors.
Examples of tasks
Example 1. Check whether the three vectors are coplanar a = {1; 2; 3}, b = {1; 1; 1}, c = {1; 2; 1}.Solution: calculate a scalar triple product of vectors
a · [b × с] = | 1 | 2 | 3 | = |
1 | 1 | 1 | ||
1 | 2 | 1 |
= 1·1·1 + 1·1·2 + 1·2·3 - 1·1·3 - 1·1·2 - 1·1·2 = 1 + 2 + 6 - 3 - 2 - 2 = 2
Answer: vectors are not coplanar as their scalar triple product is not zero.
Example 2. Prove that the three vectors a = {1; 1; 1}, b = {1; 3; 1} и c = {2; 2; 2} are coplanar. Solution: calculate a scalar triple product of vectors
a · [b × с] = | 1 | 1 | 1 | = |
1 | 3 | 1 | ||
2 | 2 | 2 |
= 1·2·3 + 1·1·2 + 1·1·2 - 1·2·3 - 1·1·2 - 1·1·2 = 6 + 2 + 2 - 6 - 2 - 2 = 0
Answer: vectors are coplanar as their scalar triple product is zero.
Example 3. Check whether the vectors are collinear a = {1; 1; 1}, b = {1; 2; 0}, c = {0; -1; 1}, d = {3; 3; 3}. Solution: Find the number of linearly independent vectors, for this we write the values of the vectors in a matrix and run at her elementary transformations 1 | 1 | 1 | ~ |
1 | 2 | 0 | |
0 | -1 | 1 | |
3 | 3 | 3 |
from 2 row we subtract the 1-th row; from 4 row we subtract the 1-th row multiplied by 3;
~ | 1 | 1 | 1 | ~ | 1 | 1 | 1 | ~ |
1 - 1 | 2 - 1 | 0 - 1 | 0 | 1 | -1 | |||
0 | -1 | 1 | 0 | -1 | 1 | |||
3 - 3 | 3 - 3 | 3 - 3 | 0 | 0 | 0 |
for 3 row add 2 row
~ | 1 | 1 | 1 | ~ | 1 | 1 | 1 |
0 | 1 | -1 | 0 | 1 | -1 | ||
0 + 0 | -1 + 1 | 1 + (-1) | 0 | 0 | 0 | ||
3 - 3 | 3 - 3 | 3 - 3 | 0 | 0 | 0 |
Since there are two non-zero row, then among the given vectors only two linearly independent vectors.
Answer: vectors are coplanar since there only two linearly independent vectors.
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Definition: Objects are coplanar if they all lie in the same plane.
Two objects are coplanar if they both lie in the same plane. In the applet above, there are 16 coplanar points. They are coplanar because they all lie in the same plane as indicated by the yellow area.
If you uncheck the 'coplanar' checkbox, the points are then randomly spread out in space and are therefore not coplanar*.
It's not just points that can be coplanar. Imagine some playing cards laying side by side on a tabletop, they are coplanar,because they both are in the same plane as each other.
In the image on the right, the two cards are both laying on a green surface. You can think of the green surface as a plane, and because the two cards are on that plane they are coplanar.
* Each time you uncheck the box a different set of random points is produced.
Parallel planes don't count
In the deck of cards on the right none of the cards are coplanar.
Each card is in a plane of its own, and although those planes are parallel to each other, that does not count as being in the same plane. So in the deck we have 52 separate (but parallel) planes with one card in each plane.
To make the cards coplanar, you would have to lay them all out on a table with no overlaps.
It only really applies to four points and up
Any set of three points are always coplanar. Put another way, you can always find a plane that passes through any set of three points. Same for a set of two points.
This is similar to the idea that in two dimensions, two points are always collinear - you can always draw a line through any two points.
It's a similar idea to colinear
Recall that points are collinear if they all lie on a straight line. Coplanar is the 3D version of this, where they all lie in the same plane.
Other point topics
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