The position vectors A and B with respect to the origin O,(-8,5) are (12,-5)and respectively. Point M is the midpoint of AB 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 term OB.
(c) Point P maps onto P' by a translation(-5,8). Give that OP = OM + 2MN, find the coordinates of
P'.

To solve this problem, we will use vector operations and properties. Let's break it down step by step.

(a) Finding the coordinates of N and M:

(i) N is the midpoint of OA. To find N, we need to calculate the average of the coordinates of O and A.

N = (O + A) / 2

The coordinates of O are (-8, 5) and the coordinates of A are (12, -5).

N = ((-8, 5) + (12, -5)) / 2
= (4, 0)

So, the coordinates of N are (4, 0).

(ii) M is the midpoint of AB. To find M, we need to calculate the average of the coordinates of A and B.

M = (A + B) / 2

The coordinates of A are (12, -5) and the coordinates of B are (-8, 5).

M = ((12, -5) + (-8, 5)) / 2
= (2, 0)

So, the coordinates of M are (2, 0).

(b) Finding the magnitude of NM:

To find the magnitude of NM, we need to find the vector NM and then calculate its magnitude.

NM = M - N

The coordinates of M are (2, 0) and the coordinates of N are (4, 0).

NM = (2, 0) - (4, 0)
= (-2, 0)

The magnitude of NM is the length of this vector, which is simply the absolute value of its x-coordinate since the y-coordinate is 0.

Magnitude of NM = |x-coordinate of NM|
= |-2|
= 2

So, the magnitude of NM is 2.

(c) Expressing vector NM in terms of vector OB:

To express vector NM in terms of vector OB, we first need to find OB.

OB = B - O

The coordinates of B are (-8, 5) and the coordinates of O are (-8, 5).

OB = (-8, 5) - (-8, 5)
= (0, 0)

Since OB is the zero vector, any vector multiplied by it will also be the zero vector.

So, vector NM in terms of vector OB is simply the zero vector.

NM = (0, 0)

(d) Finding the coordinates of P':

Given that OP = OM + 2MN, we can express this equation using vector notation:

OP = O + P'
OM = O + M
MN = M - N

Substituting the given values, we have:

O + P' = O + (O + M) + 2(M - N)

Since the origins of vectors cancel each other out, we can simplify this equation to:

P' = M + 2(M - N)

Substituting the coordinates of M and N, we have:

P' = (2, 0) + 2((2, 0) - (4, 0))
= (2, 0) + 2(-2, 0)
= (2, 0) + (-4, 0)
= (-2, 0)

So, the coordinates of P' are (-2, 0).