Question

З’ясувати, чи лежать точки А(2;–1;1), В(5;5;4), С(3;2;–1), D(1;–3;0) в одній площині?

Answer 1

Checking if Points Lie in the Same Plane

To determine if the points A(2, -1, 1), B(5, 5, 4), C(3, 2, -1), and D(1, -3, 0) lie in the same plane, we can use the concept of vectors and linear dependence.

Step 1: Find Two Vectors in the Plane

We can find two vectors in the plane formed by the points A, B, and C. Let's denote these vectors as \(\vec{AB}\) and \(\vec{AC}\).

\(\vec{AB} = B - A = (5-2, 5-(-1), 4-1) = (3, 6, 3)\)

\(\vec{AC} = C - A = (3-2, 2-(-1), -1-1) = (1, 3, -2)\)

Step 2: Check for Linear Dependence

If the vectors \(\vec{AB}\) and \(\vec{AC}\) are linearly dependent, then the points A, B, and C lie in the same plane.

The vectors \(\vec{AB}\) and \(\vec{AC}\) are linearly dependent if one is a scalar multiple of the other.

Let's calculate the cross product of \(\vec{AB}\) and \(\vec{AC}\) to check for linear dependence.

The cross product of two vectors \(\vec{u} = (u1, u2, u3)\) and \(\vec{v} = (v1, v2, v3)\) is given by: \(\vec{u} \times \vec{v} = (u2v3 - u3v2, u3v1 - u1v3, u1v2 - u2v1)\)

Calculating the cross product: \(\vec{AB} \times \vec{AC} = (6*(-2) - 3*3, 3*1 - 3*(-2), 3*3 - 6*1) = (-12 - 9, 3 + 6, 9 - 6) = (-21, 9, 3)\)

Step 3: Conclusion

If the cross product of \(\vec{AB}\) and \(\vec{AC}\) is the zero vector, then the points A, B, and C lie in the same plane.

Let's check if the cross product is the zero vector.

The cross product (-21, 9, 3) is not the zero vector, which means the points A, B, and C do not lie in the same plane.

Therefore, the points A(2, -1, 1), B(5, 5, 4), C(3, 2, -1) do not lie in the same plane.

If you have any further questions or need additional assistance, feel free to ask!