The position vectors A and B with respect to the origin O, are (-8,5)and (12,-5)respectively. Point M is the midpoint of AB and N is the midpoint of OA.
a) Find:
i) The coordinates of N and M;
Point N = OA
N = (0 + -8)/2, (0 +5)/2
= (-4, 2.5)
Point M = AB
M = (-8+12)/2, (5+ (-5)/2
= (2, 0)
ii) The magnitude of NM (3 marks)
NM = (6) (6) + (2.5) (2.5)
NM = 36 +6.25
= 42.25
= 6.5
b) Express vector NM in term OB.
c) Point P maps onto P' by a translation (-5,8) Give that
OP = OM + 2MN,
find the coordinates of P'.
(a) correct
(b) Since NM is not parallel to OB, it cannot be expressed in terms of OB alone. We do know that
ON + NM = (OB-OA)/2
OA/2 + NM = OB/2 - OA/2
NM = OB/2 - 3/2 OA
(c)
OP = OM + 2MN
= (2,0)+2(-6,2.5)
= (2,0)+(-12,5)
= (-10,5)
So, P' = (-15,13)
b) To express vector NM in terms of OB, we can use the triangle law of vector addition.
OB = B - O
= (12, -5) - (0, 0)
= (12, -5)
Now, vector NM = OB - ON
NM = OB - ON
= OB - OA
= (12, -5) - (-8, 5)
= (12 + 8, -5 - 5)
= (20, -10)
So, vector NM in terms of OB is (20, -10).
c) Given that OP = OM + 2MN, we can express this equation using vectors.
OP = OM + 2MN
OP = OM + 2(ON - OM)
OP = ON + OM
To find the coordinates of P', we can substitute the values of ON and OM.
ON = (-8,5)/2 = (-4, 2.5)
OM = (2, 0)
Substituting these values into the equation:
OP = (-4, 2.5) + (2, 0)
= (-2, 2.5)
Therefore, the coordinates of P' are (-2, 2.5).
To express vector NM in terms of OB, we need to find vector OB and then subtract vector ON from vector OM.
The coordinates of OB can be found by subtracting the coordinates of the origin O from the coordinates of point B.
OB = (12, -5) - (0, 0) = (12, -5)
Now, we can subtract vector ON from vector OM:
NM = OM - ON
NM = (-8, 5) - (0, 0) - (-4, 2.5) = (-8, 5) + (4, -2.5) = (-4, 2.5)
So, vector NM expressed in terms of OB is (-4, 2.5).
Now, to find the coordinates of point P', we can use the given formula:
OP = OM + 2MN
To find OP, we need to express vector MN and vector OM in terms of OA:
MN = ON - OM
MN = (-4, 2.5) - (-8, 5) = (-4, 2.5) + (8, -5) = (4, -2.5)
OM = OA + AM
OM = (0, 0) + (-8, 5) = (-8, 5)
Now, we can substitute these values into the formula:
OP = OA + (-8, 5) + 2(4, -2.5)
OP = (0, 0) + (-8, 5) + (8, -5)
OP = (0, 0)
Therefore, the coordinates of point P' are (0, 0).