I understand that the answer to this is a+b-c, but I'm not sure how to show working out for this. Any help would be great.

" 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.

Thanks!

AB up y axis from (0,0) s to (0,s)

then BC right s (s,s)
then CD down s to (s,0)
then DA left to (0,0)

O at (x,y)

a = -x i - y j
b = -x i + (s-y) j
c = (s-x)i (s-y) j

unknown d = (s-x)i - y j

c + a - b = (s-x)i +(s-y)j-y j -(s-y)j
= (s-x)i - y j (Caramba)
so I get
d = c + a - b

To find the vector OD, we can use the fact that the opposite sides of a square are parallel. Therefore, the vector OD will be equal to the vector OC plus the vector OB.

Given that vector OA = a, vector OB = b, and vector OC = c, we can write:

OD = OC + OB

Substituting the values of our given vectors, we have:

OD = c + b

Therefore, the vector OD in terms of a, b, and c is given by OD = c + b.

To solve this problem, we need to apply the properties of vectors and use some basic vector operations. Let's break it down step by step:

1. Recall that the vector OD can be obtained by subtracting the vector OC from the vector OA.

2. Given that OA = a and OC = c, we can write the equation as OD = OA - OC.

3. Substituting the given values, we have OD = a - c.

4. Now, we need to show that the answer is a + b - c.

5. We start by rearranging the equation. Since a + b = b + a, we can rewrite the equation as OD = (a + b) - c.

6. Next, recall that vector addition is commutative, meaning that a + b equals b + a. Therefore, we can further rewrite the equation as OD = b + a - c.

7. Finally, rearrange the terms to match the desired form, a + b - c.

So, the answer is vector OD = a + b - c, which shows the working out for obtaining the vector OD in terms of vectors a, b, and c.