Three points P(-2,1), Q(-6,4), and R(4,3) are three vertices of a parallelogram PQRS. Use your knowledge on the magnitude of vectors to find a possible coordinate for S.
PQ = Q-P = -4i+3j
QR = R-Q = 10i-j
so, since
|PS| = |QR| and PS||QR,
S-P = 10i-j
S = P+R-Q = (-2,1)+(10,-1) = (8,0)
another arrangement gives S = (0,6)
oh thank you! but what would be the other arrangement that would give (0,6), i think that's the one they're looking for
If you order the vertices PRQS you get (0,6)
Plot the points to see this.
To find a possible coordinate for point S, we need to understand the properties of a parallelogram. In a parallelogram, opposite sides are equal in length and parallel to each other. This means that the vector from P to Q is equal in length and parallel to the vector from R to S.
Let's start by finding the vector from P to Q. The coordinates of point P are (-2,1) and the coordinates of point Q are (-6,4). We can find this vector by subtracting the coordinates of point P from the coordinates of point Q:
Vector PQ = Q - P = (-6,4) - (-2,1) = (-6 - (-2), 4 - 1) = (-4, 3)
Now, we know that the vector from P to Q is (-4, 3). Since the vector from R to S is parallel to this vector, it will have the same direction but may have a different magnitude.
To find a possible coordinate for point S, we can start with the coordinates of point R (4,3) and add the vector PQ to it:
Coordinates of S = Coordinates of R + Vector PQ = (4,3) + (-4,3) = (4 - 4, 3 + 3) = (0, 6)
Therefore, a possible coordinate for point S is (0, 6).