Let A and B be any two points on the plane and O be the origin. Prove that vector AB = vector OB - vector OA.
Let x be any other point on the plane prove that vector AB = vector xB - vector xA.
To prove that vector AB = vector OB - vector OA, we need to use the definition of vector subtraction. Vector subtraction can be done by reversing the direction of the vector being subtracted and then adding it to the other vector. Here's how you can prove it:
1. Start with the definition of vector subtraction:
Vector AB = Vector OB - Vector OA
2. Express vector OB and vector OA in terms of their components:
Vector OB = (x_B, y_B)
Vector OA = (x_A, y_A)
3. Substitute the components into the equation:
Vector AB = (x_B, y_B) - (x_A, y_A)
4. Apply the definition of vector subtraction:
Vector AB = (x_B - x_A, y_B - y_A)
5. We can conclude that vector AB is equal to the difference of the corresponding components:
Vector AB = (x_B - x_A, y_B - y_A)
This shows that vector AB can be expressed as the difference of the corresponding components of vector OB and vector OA, proving the given statement.
Now, let's prove that vector AB = vector xB - vector xA, where x is any other point on the plane using the same approach:
1. Start with the definition of vector subtraction:
Vector AB = Vector xB - Vector xA
2. Express vector xB and vector xA in terms of their components:
Vector xB = (x_B, y_B)
Vector xA = (x_A, y_A)
3. Substitute the components into the equation:
Vector AB = (x_B, y_B) - (x_A, y_A)
4. Apply the definition of vector subtraction:
Vector AB = (x_B - x_A, y_B - y_A)
5. We can conclude that vector AB is equal to the difference of the corresponding components:
Vector AB = (x_B - x_A, y_B - y_A)
This shows that vector AB can be expressed as the difference of the corresponding components of vector xB and vector xA, proving the given statement for any point x on the plane.