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’.
Where does the 1 on N come from?
A(-8,5), N(x1,y1), O(0,0), M(x2,y2), B(12,-5).
a. Point N:
x1-(-8) = 0-x1.
2x1 = -8.
x1 = -4.
y1-5 = 0-y.
2y1 = 5.
Y1 = 2.5.
Point M:
x2-0 = 12-x2.
2x2 = 12.
X2 = 6.
y2-0 = -5-y2.
2y2 = -5
Y2 = -2.5
(NM)^2 = (x2-x1)^2 + (y2-y1)^2 =
(6-(-4))^2 + (-2.5-2.5)^2 = 100 + 25 =
125.
NM = 11.18.
b. (OB)^2 = (X2)^2 + (Y2)^2 = 36 + 6.25 = 42.25.
OB = 6.5.
NM/OB = 11.18/6.5.
NM = 1.72*OB.
(a)
(i) To find the coordinates of N, we need to find the midpoint of OA.
The coordinates of A are (-8, 5) and the coordinates of O are (0, 0).
Using the midpoint formula, we can calculate the coordinates of N as follows:
x-coordinate of N = (x-coordinate of A + x-coordinate of O) / 2 = (-8 + 0) / 2 = -4
y-coordinate of N = (y-coordinate of A + y-coordinate of O) / 2 = (5 + 0) / 2 = 2.5
Therefore, the coordinates of N are (-4, 2.5).
To find the coordinates of M, we need to find the midpoint of B.
The coordinates of B are (12, -5).
Using the midpoint formula, we can calculate the coordinates of M as follows:
x-coordinate of M = (x-coordinate of B + x-coordinate of O) / 2 = (12 + 0) / 2 = 6
y-coordinate of M = (y-coordinate of B + y-coordinate of O) / 2 = (-5 + 0) / 2 = -2.5
Therefore, the coordinates of M are (6, -2.5).
(ii) To find the magnitude of NM, we can use the distance formula:
Magnitude of NM = sqrt((x-coordinate of M - x-coordinate of N)^2 + (y-coordinate of M - y-coordinate of N)^2)
= sqrt((6 - (-4))^2 + (-2.5 - 2.5)^2)
= sqrt(10^2 + 0^2)
= sqrt(100)
= 10
Therefore, the magnitude of NM is 10.
(b) To express vector NM in terms of OB, we can use the triangle law of vector addition:
NM = OB - ON
Since we know the coordinates of N and M, and the position vector OB is given by the difference of the coordinates of B and O, we can substitute these values and calculate NM as follows:
NM = (12, -5) - (-4, 2.5)
= (12 + 4, -5 - 2.5)
= (16, -7.5)
Therefore, vector NM can be written as (16, -7.5).
(c) Given that OP = OM + 2MN, we can rearrange this equation to find the coordinates of P' as follows:
OP = OM + 2MN
OP = (12, -5) + 2(-4, 2.5)
OP = (12, -5) + (-8, 5)
OP = (12 - 8, -5 + 5)
OP = (4, 0)
Since OP = OP' + (-5, 8) (translation vector), we can set up the equation:
OP' + (-5, 8) = (4, 0)
Solving for P', we get:
OP' = (4, 0) - (-5, 8)
= (4 + 5, 0 - 8)
= (9, -8)
Therefore, the coordinates of P' are (9, -8).
(a) (i) To find the coordinates of N, we need to find the midpoint of OA. The midpoint formula states that the coordinates of the midpoint of a line segment with endpoints (x₁, y₁) and (x₂, y₂) are given by the average of the coordinates:
N = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)
In this case, the coordinates of A are (-8, 5) and the coordinates of O (the origin) are (0, 0). Substituting these values into the formula, we have:
N = ((-8 + 0) / 2, (5 + 0) / 2)
= (-4, 2.5)
Therefore, the coordinates of N are (-4, 2.5).
To find the coordinates of M, we need to find the midpoint of B, which has coordinates (12, -5). Since B is already the midpoint of OB, the coordinates of M are the same as the coordinates of B:
M = (12, -5)
(ii) To find the magnitude of NM, we can use the distance formula, which states that the distance between two points (x₁, y₁) and (x₂, y₂) is given by:
d = √((x₂ - x₁)² + (y₂ - y₁)²)
In this case, the coordinates of N are (-4, 2.5) and the coordinates of M are (12, -5). Substituting these values into the formula, we have:
NM = √((12 - (-4))² + (-5 - 2.5)²)
= √((12 + 4)² + (-5 - 2.5)²)
= √(16² + (-7.5)²)
= √(256 + 56.25)
= √(312.25)
≈ 17.67
Therefore, the magnitude of NM is approximately 17.67.
(b) To express vector NM in terms of OB, we need to find the difference between the coordinates of N and M. The difference between two points (x₂, y₂) and (x₁, y₁) is given by subtracting the corresponding coordinates:
NM = (x₂ - x₁, y₂ - y₁)
In this case, the coordinates of N are (-4, 2.5) and the coordinates of M are (12, -5). Substituting these values into the formula, we have:
NM = (12 - (-4), -5 - 2.5)
= (16, -7.5)
Therefore, vector NM can be expressed as (16, -7.5) in terms of OB.
(c) Given that OP = OM + 2MN, and we know the coordinates of O (0, 0), M (12, -5), and N (-4, 2.5), we can substitute these values into the equation to find the coordinates of P':
OP = OM + 2MN
P - O = M - O + 2(N - M)
P = M + 2(N - M) + O
P = M + 2N - 2M
P = -M + 2N + M
Substituting the coordinates of M (12, -5) and N (-4, 2.5) into the equation, we have:
P = (-12, 5) + 2(-4, 2.5) + (12, -5)
P = (-12, 5) + (-8, 5) + (12, -5)
P = (-20, 10)
Therefore, the coordinates of P' are (-20, 10).