Question

O is any point in the plane of the square ABCD. The vectors

OA, OB and OC are a, b and c respectively. Find the vector
OD in terms of a, b and c.

Answer 1

The vector OD can be expressed in terms of a, b, and c as b - 1/2 c.

Explanation:

Since ABCD is a square, we know that AB = BC = CD = DA, and the diagonals AC and BD are perpendicular and bisect each other. Let M be the midpoint of AC, then OM is perpendicular to AC and BM, which means that OM is parallel to AD and BC.

Since OM is parallel to AD, we can write:

OD = OA + AD

And since OM is parallel to BC, we can write:

OC = OB + BM

But since M is the midpoint of AC, we have:

BM = MC = 1/2 AC

Therefore, we can substitute OM = 1/2 AC for BM in the second equation to get:

OC = OB + 1/2 AC

Now we can substitute the given vectors for OA, OB, and OC to get:

OD = OA + AD = OA + OC - AC

OD = a + (b + 1/2 c) - (a + c)

OD = b - 1/2 c

Therefore, the vector OD can be expressed in terms of a, b, and c as b - 1/2 c.