Four points are always coplanar if

Four points are always  coplanar if triple scalar  product is zero.

Step-by-step explanation:

We have four points and we find the condition for four points are coplanar.

Let A,B,C and D are four points .We have to find Vector AB, AC and AD.

Then we find triple scalar product of  vectors AB,AC and AD.

Triple scalar product=

To find triple scalar product we use determinant

Suppose A(1,0,-1),B(0,2,3),C(-2,1,1) and D(4,2,3) are four points .

We shall prove that A,B,C and D are coplanar.

We find vector AB,AC and AD

=Coordinate of B -coordinate of A=

=Coordinate of C-coordinate of A=

=Coordinate of D- coordinate of A=

Now , we find triple scalar product

Expand alon

=-1(4-4)-2(-12-6)+4(-6-3)

=0+36-36=0

Hence, triple scalar product is zero therefore, four points A,B,C and D are coplanar.

Four points are always coplanar if triple scalar product is zero.

1) is right

step-by-step explanation:

when rotating a figure 180 degrees, imagine that you are able to take the figure and turn it clockwise or counterclockwise around a center point. to rotate a figure 180 degrees, you will need to apply the rule (x, y) → (-x, -y). start by using a coordinate grid with coordinates for each vertex of the figure. the center point of the coordinate grid is located at (0, 0), which is what you will rotate the figure around. write down the original coordinates of the shape you are going to rotate. then, apply the rule. for example, coordinate (1, 2) becomes (-1, -2). another example would be, coordinate (-4, -2) becoming (4, 2). after you change each original coordinate to the rotated coordinate, you will draw your new figure.

step-by-step explanation: