A parallelogram is formed by R^3 by the vectors PA=(3,2,-3) and PB=(4,1,5). The point P=(0,2,3).
a) what are the location of the vertices?
b) what are the vectors representing the diagonals?
c) what are the length of the diagonals?
(a)
Parallelogram: PACB, where
P(0,2,3) is a vertex.
A=P+PA=(0,2,3)+(3,2,-3)=A(3,4,0)
B=P+PB=(0,2,3)+(4,1,5)=B(4,3,8)
C=A+PB=(3,4,0)+(4,1,5)=C(7,5,5)
(b)
PC=C-P=(7,5,5)-(0,2,3)=(7,3,2)
AB=B-A=(4,3,8)-(3,4,0)=(1,-1,8)
Lengths
PC=sqrt(7²+3²+2²)=sqrt(49+9+4)=√62
I leave you to find the length of AB in a similar way to finding length of PC.
To find the vertices of the parallelogram formed by the given vectors, we need to start with the point P and find the other three vertices by adding the given vectors to P.
a) To find the location of the vertices:
1. Start with the point P=(0,2,3).
2. Add vector PA=(3,2,-3) to P to find the first vertex:
P + PA = (0,2,3) + (3,2,-3) = (3,4,0)
3. Add vector PB=(4,1,5) to P to find the second vertex:
P + PB = (0,2,3) + (4,1,5) = (4,3,8)
4. Subtract vector PA from P to find the third vertex:
P - PA = (0,2,3) - (3,2,-3) = (-3,0,6)
Therefore, the vertices of the parallelogram are:
P1 = (0,2,3)
P2 = (3,4,0)
P3 = (4,3,8)
P4 = (-3,0,6)
b) To find the vectors representing the diagonals:
1. The first diagonal is the line segment connecting the vertices P2 and P4. Let's call this vector D1.
D1 = P4 - P2 = (-3,0,6) - (3,4,0) = (-6,-4,6)
2. The second diagonal is the line segment connecting the vertices P1 and P3. Let's call this vector D2.
D2 = P3 - P1 = (4,3,8) - (0,2,3) = (4,1,5)
Therefore, the vectors representing the diagonals are:
D1 = (-6,-4,6)
D2 = (4,1,5)
c) To find the length of the diagonals, we can use the formula for the magnitude of a vector:
1. Calculate the magnitude of vector D1:
|D1| = sqrt((-6)^2 + (-4)^2 + 6^2) = sqrt(36 + 16 + 36) = sqrt(88)
2. Calculate the magnitude of vector D2:
|D2| = sqrt(4^2 + 1^2 + 5^2) = sqrt(16 + 1 + 25) = sqrt(42)
Therefore, the length of the diagonals are:
|D1| = sqrt(88)
|D2| = sqrt(42)
To find the answers to the questions, we need to understand the properties of a parallelogram formed by two vectors in R^3.
a) To find the locations of the vertices of the parallelogram, we need to add the given vectors to the point P.
First, add vector PA to point P:
P + PA = (0,2,3) + (3,2,-3) = (3,4,0)
Next, add vector PB to point P:
P + PB = (0,2,3) + (4,1,5) = (4,3,8)
The locations of the vertices of the parallelogram are P(0,2,3), A(3,4,0), B(4,3,8).
b) The vectors representing the diagonals of a parallelogram can be found by subtracting the corresponding vertices.
The diagonal vectors are:
AC = C - A = (4,3,8) - (3,4,0) = (1,-1,8)
BD = D - B = (3,4,0) - (4,3,8) = (-1,1,-8)
c) The length of the diagonals can be calculated using the distance formula, ||AC|| and ||BD||.
||AC|| = √((1)^2 + (-1)^2 + (8)^2) = √(1 + 1 + 64) = √66
||BD|| = √((-1)^2 + (1)^2 + (-8)^2) = √(1 + 1 + 64) = √66
Therefore, the length of both diagonals ||AC|| and ||BD|| is √66.