Two points P and Q have coordinates (-2, 3) and (1, 3) respectively. A translation map point P to P’ (10, 10)
(a) Find the coordinates of Q’ the image of Q under the translation
(b) The position vector of P and Q in (a) above are p and q respectively given that mp – nq =[-12,9 ]. find the value of m and n
Rule for translation:
(x,y)-> (x+h, y+k)
Solve for h,k from
(-2,3)->(10,10)
(a) hence find the image Q' of Q.
(b)
if mp-nq=(-12,9), this means
m(-2,3)-n(1,3)=(-12,9)
Split into individual equations for x and y to get
-2m-n = -12
3m-3n = 9
Solve for m,n.
(_12,9)
(a) To find the image of point Q, we need to apply the same translation to the coordinates of Q as we did for point P.
Given that point P was translated to P' with coordinates (10, 10), we can find the translation vector by subtracting the coordinates of point P from point P':
Translation vector = P' - P = (10, 10) - (-2, 3) = (10 + 2, 10 - 3) = (12, 7)
Now, to find the coordinates of Q', we need to add the translation vector to the coordinates of Q:
Coordinates of Q' = Coordinates of Q + Translation vector = (1, 3) + (12, 7) = (1 + 12, 3 + 7) = (13, 10)
Therefore, the coordinates of Q' are (13, 10).
(b) Let the position vectors of P and Q be p and q respectively. Given that mp - nq = [-12, 9], we can denote p = (p1, p2) and q = (q1, q2). So we have the following equations:
mp1 - nq1 = -12 ----------- (1)
mp2 - nq2 = 9 ----------- (2)
Using the coordinates of P and Q, we can write:
-2m - n = -12 ----------- (3)
3m - 3n = 9 ----------- (4)
Now, we need to solve equations (3) and (4) simultaneously to find the corresponding values of m and n.
Multiplying equation (3) by 3 and equation (4) by 2, we get:
-6m - 3n = -36 ----------- (5)
6m - 6n = 18 ----------- (6)
Adding equations (5) and (6) eliminates the variable m:
-9n = -18
Dividing both sides of the equation by -9:
n = 2
Substituting the value of n = 2 into equation (3):
-2m - 2 = -12
Adding 2 to both sides of the equation:
-2m = -10
Dividing both sides of the equation by -2:
m = 5
Therefore, the values of m and n are m = 5 and n = 2.