Find the coordinates of X with respect to the vectors A, B and C
X = (1, 0, 0), A = (1, 1, 1), B = (-1, 1, 0), C = (1, 0, -1)
X = (1, 1, 1) , A = (0, 1, -1), B = (1, 1, 0), C = (1, 0, 2)
To find the coordinates of X with respect to the vectors A, B, and C, we need to express X as a linear combination of A, B, and C.
For the first set of vectors:
X = 1A + 0B + 0C
X = A
The coordinates of X with respect to A, B, and C are (1, 0, 0).
For the second set of vectors:
X = 1A + 1B + 1C
X = (1, 1, 1)
The coordinates of X with respect to A, B, and C are (1, 1, 1).
To find the coordinates of X with respect to the vectors A, B, and C, we can use the method of projection.
Step 1: Setting up the coordinate system:
Let's assume that the vectors A, B, and C form the basis of a coordinate system. In other words, any point in this coordinate system can be represented as a linear combination of A, B, and C.
Step 2: Writing the given coordinates of X as a linear combination of A, B, and C:
For the first set of coordinates, X = (1, 0, 0), we can write this as:
X = a * A + b * B + c * C
where a, b, and c are the coefficients that we need to find.
Step 3: Solving for the coefficients:
Substituting the given vectors and coordinates into the above equation, we have:
(1, 0, 0) = a * (1, 1, 1) + b * (-1, 1, 0) + c * (1, 0, -1)
Expanding this equation, we get:
(1, 0, 0) = (a - b + c, a + b, a - c)
Comparing the components on both sides of the equation, we can set up a system of linear equations:
1 = a - b + c
0 = a + b
0 = a - c
Using any method to solve this system of equations, we find that a = 0.5, b = -0.5, and c = 0.5.
Therefore, the coordinates of X with respect to the vectors A, B, and C are (0.5, -0.5, 0.5).
For the second set of coordinates, X = (1, 1, 1), and the vectors A, B, and C are given as A = (0, 1, -1), B = (1, 1, 0), and C = (1, 0, 2). Following the same steps as above, we can obtain the coordinates of X with respect to A, B, and C.