The position vectors of points A and B with respect to the origin 0, are (-8,5) and (12,-5) respectively.
Point M is the midpoint of B and N is the midpoint of OA.
(a) Find:
(i) The coordinates of N and M
(ii) The magnitude of NM
(b) Express vector NM in terms of OB
(c) Point P maps onto P’ by a translation (-5,8) . Given that
OP = OM + 2MN, find the coordinates of p’.
To solve this problem, we will use vector operations and formulas. Let's break it down step by step:
(a) (i) To find the coordinates of N, we need the midpoint of OA. The midpoint of a segment is the average of the coordinates of its endpoints. So, to find N, we can use the formula:
N = (O + A) / 2
Since the origin is at (0,0), we have:
N = (0,0 + -8,5) / 2
N = (-8,5) / 2
N = (-4,25)
Similarly, to find the coordinates of M, we need the midpoint of B. Applying the same formula:
M = (B + O) / 2
M = (12,-5 + 0,0) / 2
M = (12,-5) / 2
M = (6,-2.5)
(ii) To find the magnitude of NM, we can use the distance formula:
Magnitude of NM = √[(x2 - x1)^2 + (y2 - y1)^2]
Substituting the coordinates, we get:
Magnitude of NM = √[(6 - (-4))^2 + (-2.5 - 2)^2]
Magnitude of NM = √[10^2 + (-4.5)^2]
Magnitude of NM = √[100 + 20.25]
Magnitude of NM = √120.25
Magnitude of NM = 10.97 (rounded to two decimal places)
(b) To express vector NM in terms of OB, we can use the tip-to-tail method. We subtract the initial point of a vector from the final point:
NM = OB - ON
Substituting the values:
NM = (12,-5) - (-4,25)
NM = (12 + 4,-5 - 25)
NM = (16,-30)
So, vector NM can be expressed as (16,-30) in terms of OB.
(c) From the given information, we are told that OP = OM + 2MN. Let's substitute the coordinates and solve for P’.
OP = (0,0) + 2MN
P = 2MN
Since we already found the coordinates for M and N, we can substitute them into the equation:
P = 2(-4,25, -2.5)
P = (-8.5, -5)
To find P’, we will translate P by (-5,8):
P’ = P + (-5,8)
P’ = (-8.5, -5) + (-5,8)
P’ = (-8.5 - 5, -5 + 8)
P’ = (-13.5, 3)
So, the coordinates of P’ are (-13.5, 3).