QP is (x y) = (3 -1) + r (14 10)
QR is (x y) = (3 -1) + s (17 -9)
PR is (x y) = (0 18) + t (5 -7)
where r, s, and t are scalars.
1. Use vector methods to find the coordinates of P, Q, and R.
2. Find vectors PQ and PR and evaluate PQ*PR
3. Find size of QPR
4. Find the area of triangle PQR
To find the coordinates of points P, Q, and R, we can use the given vector equations:
1. Coordinates of P:
The vector equation for P is given as:
(x, y) = (3, -1) + r(14, 10)
So, the coordinates of P are (3 + 14r, -1 + 10r).
2. Coordinates of Q:
The vector equation for Q is given as:
(x, y) = (3, -1) + s(17, -9)
So, the coordinates of Q are (3 + 17s, -1 - 9s).
3. Coordinates of R:
The vector equation for R is given as:
(x, y) = (0, 18) + t(5, -7)
So, the coordinates of R are (0 + 5t, 18 - 7t).
Now let's move on to the next parts of the question:
2. Vectors PQ and PR:
Vector PQ is obtained by subtracting the coordinates of Q from the coordinates of P:
PQ = (3 + 14r, -1 + 10r) - (3 + 17s, -1 - 9s)
= (14r - 17s, 10r + 9s)
Vector PR is obtained by subtracting the coordinates of R from the coordinates of P:
PR = (3 + 14r, -1 + 10r) - (0 + 5t, 18 - 7t)
= (3 + 14r - 5t, -1 + 10r + 7t)
Evaluating PQ dot PR:
PQ dot PR = (14r - 17s)(3 + 14r - 5t) + (10r + 9s)(-1 + 10r + 7t)
3. Size of QPR (Magnitude of PQ cross PR):
The size of QPR can be found by taking the cross product of vectors PQ and PR and finding its magnitude:
|PQ cross PR| = |(14r - 17s, 10r + 9s, 0)|
4. Area of triangle PQR:
The area of triangle PQR can also be found using the magnitude of the cross product of vectors PQ and PR:
Area = 1/2 |PQ cross PR|