Determine the Value of k for a Point to Lie on the Same Plane as Given Points

Seamlessly determining if a point lies on the same plane as other given points involves a series of calculations that leverage vector operations. This technique is particularly useful in fields such as computer graphics, robotics, and engineering. In this article, we will guide you through the process of finding the value of k for which the point D (1, k, -2) lies on the same plane as the points A (4, -2, 6), B (0, 1, 0), and C (1, 0, -5).

Understanding the Problem

The problem boils down to finding k such that the point D lies on the same plane as points A, B, and C. The normal vector of this plane can be determined using the cross product of vectors A - B and C - B. Once we have this normal vector, we need to ensure that vector D - B is perpendicular to it.

Step-by-Step Solution

To solve this problem, we will follow a systematic approach:

Find the Vectors: Calculate the vectors A - B, C - B, and D - B. These will serve as the building blocks for our calculations.

A - B  (4 - 0, -2 - 1, 6 - 0)  (4, -3, 6)C - B  (1 - 0, 0 - 1, -5 - 0)  (1, -1, -5)D - B  (1 - 0, k - 1, -2 - 0)  (1, k - 1, -2)

Find the Cross Product: The cross product of vectors A - B and C - B gives us the normal vector of the plane.

Using the determinant method:

(A - B) times; (C - B)  begin{vmatrix}mathbf{i}  mathbf{j}  mathbf{k} 4  -3  6 1  -1  -5 end{vmatrix} mathbf{i}((-3)(-5) - (6)(-1)) - mathbf{j}((4)(-5) - (6)(1))   mathbf{k}((4)(-1) - (-3)(1)) mathbf{i}(15   6) - mathbf{j}(-20 - 6)   mathbf{k}(-4   3) 21mathbf{i}   26mathbf{j} - mathbf{k}

The resulting normal vector is (21, 26, -1).

Check Perpendicularity: To ensure that vector D - B is perpendicular to the normal vector, their dot product must be zero.

(D - B) middot; (A - B) times; (C - B)  (1, k - 1, -2) middot; (21, 26, -1)  0

This simplifies to:

21   26(k - 1)   2  021   26k - 26   2  026k - 3  026k  3k  frac{3}{26}

Hence, the value of k is frac{3}{26}.

Conclusion

By following these steps, we have determined that the point D (1, k, -2) with k frac{3}{26} lies on the same plane as points A (4, -2, 6), B (0, 1, 0), and C (1, 0, -5). This calculation ensures that the vector D - B is perpendicular to the normal vector of the plane defined by A - B and C - B.

For further verification, we can check if the vectors A - B, C - B, and D - B are linearly dependent. However, this is not necessary in the context of the problem as we have already shown that the conditions for perpendicularity are met.