Find the position vector of the point D such that ABCD is a parallelogram. I used AB=DC where AB and DC are ve

Question

Find the position vector of the point D such that ABCD is a parallelogram. I used AB=DC where AB and DC are vectors and AB=DC in terms of their magnitude but couldn;t find a valuse for p.

OA= 2i + 3j - 4k
OB= 5i - j + 2k
OC= 11i + pj +14k

Can anyone help me see if p can be found through the magnitude thing? cause i got p^2 + 8p +83=0 when i used AB=DC but the equation can not be solved.

Answer 1

BD = BA+BC
OD = BD-OB = OA+BC = 8i+(p+4)j+8k

p can be any number because you only know two vertices A and B, the third vertex C is free to pick.

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bobqwatso,
A, B, C, and D are vertices, not vectors.

Answer 2

Let A and B be vectors, and let D = A + B. Then D is the last vertex of the parallelogram whose first three points are 0, A, and B.

To solve your question, let's translate C to the origin by subtracting 0C from 0A and 0B. Then the last vertex of the parallelogram will be
D = 0C + (0A - 0C) + (0B - 0C) = 0A + 0B - 0C
D = -4i + (2-p)j - 16k