Find the coordinates of X with respect to the vectors A, B and C
A)X = (1, 0, 0), A = (1, 1, 1), B = (-1, 1, 0), C = (1, 0, -1)
B)X = (1, 1, 1) , A = (0, 1, -1), B = (1, 1, 0), C = (1, 0, 2)
A) To find the coordinates of X with respect to vectors A, B, and C, we need to express X as a linear combination of A, B, and C.
Let's try to solve for the coefficients x, y, and z in the equation X = xA + yB + zC.
Substituting the given values:
(1, 0, 0) = x(1, 1, 1) + y(-1, 1, 0) + z(1, 0, -1)
Expanding the equation:
(1, 0, 0) = (x, x, x) + (-y, y, 0) + (z, 0, -z)
Combining like terms:
(1, 0, 0) = (x - y + z, x + y, x - z)
Setting the corresponding components equal to each other:
1 = x - y + z
0 = x + y
0 = x - z
From the second and third equations, we can solve for x and z in terms of y:
x = -y
z = x
Substituting these expressions back into the first equation:
1 = -y - y + y
1 = -y
Therefore, the coordinates of X with respect to the vectors A, B, and C are (-1, 0, 1).
B) Let's proceed similarly to the previous example.
(1, 1, 1) = x(0, 1, -1) + y(1, 1, 0) + z(1, 0, 2)
(1, 1, 1) = (0, x, -x) + (y, y, 0) + (z, 0, 2z)
(1, 1, 1) = (y + z, x + y, -x + 2z)
1 = y + z
1 = x + y
1 = -x + 2z
From the second equation, we can solve for x in terms of y:
x = 1 - y
Substituting this expression back into the third equation:
1 = -(1 - y) + 2z
1 = -1 + y + 2z
2 = y + 2z
Considering the first equation, we know that:
1 = y + z
We can solve these two equations to find the values of y and z:
y = 1 - z
Substituting this expression back into the previous equation:
2 = 1 - z + 2z
1 = z
Therefore, the coordinates of X with respect to the vectors A, B, and C are (0, 1, 1).
To find the coordinates of X with respect to the vectors A, B, and C, we can use the matrix representation method. We will set up a system of equations and solve for the coefficients of A, B, and C that result in X.
A) X = (1, 0, 0), A = (1, 1, 1), B = (-1, 1, 0), C = (1, 0, -1):
Let's represent X using the coefficients x, y, and z:
X = xA + yB + zC
Substituting the given values:
(1, 0, 0) = x(1, 1, 1) + y(-1, 1, 0) + z(1, 0, -1)
This forms the following system of equations:
x - y + z = 1
x + y = 0
x + z = 0
Solving this system of equations, we find:
x = -1/2
y = 1/2
z = 1/2
Therefore, the coordinates of X with respect to the vectors A, B, and C are (-1/2, 1/2, 1/2).
B) X = (1, 1, 1), A = (0, 1, -1), B = (1, 1, 0), C = (1, 0, 2):
Let's represent X using the coefficients x, y, and z:
X = xA + yB + zC
Substituting the given values:
(1, 1, 1) = x(0, 1, -1) + y(1, 1, 0) + z(1, 0, 2)
This forms the following system of equations:
y + z = 1
x + y + z = 1
-x + z = 1
Solving this system of equations, we find:
x = 2
y = -1
z = 2
Therefore, the coordinates of X with respect to the vectors A, B, and C are (2, -1, 2).