Answer:(a) To find the position vectors PO and PR, we need to subtract the position vectors of the vertices P, O, and R.
Given that OQ = OR = OP, we can find the position vector of O by taking the average of OQ and OR.
O = (OQ + OR) / 2
Substituting the values, we get:
O = (3 + 3) / 2 = 6 / 2 = 3
Now, we can find PO and PR by subtracting the position vectors:
PO = P - O
PR = R - P
Substituting the values, we get:
PO = P - 3
PR = R - P
(b) To find cos RPQ, we can use the dot product formula:
cos RPQ = (PQ · PR) / (|PQ| * |PR|)
The dot product of PQ and PR is given by:
PQ · PR = (Q - P) · (R - P)
Substituting the values, we get:
PQ · PR = (3 - P) · (3 - P)
Now, to find cos RPQ, we need to calculate the magnitudes of PQ and PR:
|PQ| = |Q - P|
|PR| = |R - P|
Substituting the values, we get:
|PQ| = |3 - P|
|PR| = |3 - P|
Therefore, cos RPQ = ((3 - P) · (3 - P)) / (|3 - P| * |3 - P|).
(c) To find sin RPQ, we can use the cross product formula:
sin RPQ = |PQ x PR| / (|PQ| * |PR|)
The cross product of PQ and PR is given by:
PQ x PR = (Q - P) x (R - P)
Substituting the values, we get:
PQ x PR = (3 - P) x (3 - P)
Now, to find sin RPQ, we need to calculate the magnitudes of PQ and PR:
|PQ| = |Q - P|
|PR| = |R - P|
Substituting the values, we get:
|PQ| = |3 - P|
|PR| = |3 - P|
Therefore, sin RPQ = |(3 - P) x (3 - P)| / (|3 - P| * |3 - P|).
To find the area of triangle PQR, we can use the formula:
Area = (1/2) * |PQ x PR|
Substituting the values, we get:
Area = (1/2) * |(3 - P) x (3 - P)|
Finally, we can simplify the expression and evaluate the result to obtain the area of triangle PQR in the required form, a3.
Step-by-step explanation:
