If P is at (0, 0), then Q would be at (2, 5), and R would be at (6, 15). You can choose different values for
to find other sets of coordinates for P, Q, and R.
To find the coordinates of points P, Q, and R, given the vector representations of PQ and PR as 2a + 5b and 6a + 15b respectively, we need to express the coordinates of these points in terms of a and b. Let's do this step by step:
1. Start with point P. Let's denote the coordinates of P as
We can express the vector PQ as follows:
Since we don't have information about c, we can assume it's zero unless specified otherwise. So, we'll focus on the first two components:
Now, we can equate the coefficients of a and b to find
Equating coefficients of a:
Equating coefficients of b:
2. Next, let's move on to point R. Denote the coordinates of R as
. We can express the vector PR as follows:
Again, we'll focus on the first two components:
Equating coefficients of a:
Equating coefficients of b:
3. Now, we have three equations (equations 1, 2, and 3) to solve for
From equation 1:
From equation 2:
4. Next, we'll use equations 3 and 4 to solve for
From equation 3:
From equation 4: x_r = 6 + x_p
5. At this point, we have expressions for
. Now, you can choose arbitrary values for
, and you can calculate the corresponding values for
using the expressions derived in steps 3 and 4.
For example, let's say you choose
From step 3:
From step 4:
