Final answer:
The value of k for which the vector from A to B is perpendicular to the vector from A to P is found by using the dot product of the vectors, which must equal zero. After setting up the equation and computing the dot product of (10, 3, -9) and (k + 5, k, k - 4), we solve for k to find the required value.
Step-by-step explanation:
To find the value of k for which the vector from point A to point B is perpendicular to the vector from point A to point P, we must use the concept of dot product of two vectors. The dot product of two perpendicular vectors is zero. Therefore, we can set up the following equation using the vector coordinates:
AB ⋅ AP = 0
Where AB is the vector from A to B, and AP is the vector from A to P. The components of AB are obtained by subtracting the coordinates of A from the coordinates of B, resulting in:
AB = (5 - (-5), 3 - 0, -5 - 4) = (10, 3, -9)
AP is represented by the coordinates of P minus the coordinates of A:
AP = (k - (-5), k - 0, k - 4) = (k + 5, k, k - 4)
Determining the dot product gives us:
(10, 3, -9) ⋅ (k + 5, k, k - 4) = 10(k + 5) + 3k - 9(k - 4) = 0