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
Your notation is slightly different from the one I am used to.
QP is a line with (x,y) = (3,-1) + r(14,10)
or
x = 3 + 14r
y = -1 + 10r
PR:
x = 0 + 5t
y = 18 - 7t
P is the intersection of QP and PR
so
3 + 14r = 0 + 5t ----> 14r - 5t = -3 **
-1 + 10r = 18 - 7t ---> 10r + 7t = 19 ***
** times 7 ----> 98r - 35t = -21
*** times 5 ---> 50r + 35t = 95
add them:
148r = 74
r = 1/2
then in
x = 3 + 14r -----> x = 3 + 7 = 10
y = -1 + 10r ----> y = -1 + 5 = 4
P is (10,4)
Find Q and R the same way.
(I bet Q is (3,-1) , how do you think I found that without any work ??)
#2, I will do vector PQ
vector PQ = [3-10, -1-4] or [-7,-5]
if PQ*PR is the 'dot' product, then you should have a method in your text or notes, it is one of the easiest calculations
(in my notation, we use [....] for vectors and (...) for points )
#3 I don't know what you mean by "size" of QPR, to me that would mean area.
In that case what is the difference between #3 and #4 ?
To solve this problem, we will use the given vector equations and perform vector operations to find the coordinates of the points, vectors PQ and PR, evaluate their dot product, find the size of QPR, and finally find the area of triangle PQR.
1. Finding the coordinates of P, Q, and R:
We are given the vector equations for P, Q, and R:
QP: (x, y) = (3, -1) + r(14, 10)
QR: (x, y) = (3, -1) + s(17, -9)
PR: (x, y) = (0, 18) + t(5, -7)
To find the coordinates, we substitute different values of r, s, and t into these equations:
For P (QP):
When r = 0, (x, y) = (3, -1) + 0(14, 10) = (3, -1)
Hence, the coordinates of point P are (3, -1).
For Q (QR):
When s = 0, (x, y) = (3, -1) + 0(17, -9) = (3, -1)
Hence, the coordinates of point Q are (3, -1).
For R (PR):
When t = 0, (x, y) = (0, 18) + 0(5, -7) = (0, 18)
Hence, the coordinates of point R are (0, 18).
2. Finding vectors PQ and PR, and evaluating PQ*PR:
We can find vectors PQ and PR by subtracting the coordinates of one point from the other.
PQ = Q - P = (3, -1) - (3, -1) = (0, 0)
PR = R - P = (0, 18) - (3, -1) = (-3, 19)
To evaluate PQ*PR, we take the dot product of these two vectors:
PQ*PR = (0, 0) ⋅ (-3, 19) = (0 * -3) + (0 * 19) = 0
3. Finding the size of QPR:
The size (magnitude) of vector QPR is given by the formula ||QPR|| = √(x^2 + y^2), where x and y are the components of the vector.
QPR = QR - PR = (3, -1) + s(17, -9) - (0, 18) - t(5, -7)
= (3 + 17s - 5t, -1 - 9s + 7t)
Using the formula, the size of QPR = √((3 + 17s - 5t)^2 + (-1 - 9s + 7t)^2)
4. Finding the area of triangle PQR:
The area of a triangle can be calculated as half of the magnitude of the cross product of two of its sides. In this case, we can use vectors PQ and PR.
To find the cross product of two vectors PQ × PR, we find the determinant of the 2x2 matrix formed by the components of these vectors:
PQ × PR = |(0, 0) (-3, 19)|
|(0, 0) (17, -9)|
Taking the determinant, we have:
PQ × PR = (0 * -9) - (0 * 17) = 0
Finally, the area of triangle PQR = 0.