let vector U = (vector u1, vector u2) vector V = (vector v1, vector v2) and vector W = (vector w1, vector w2) Prove each property using Cartesian vectors:
a) (vector U+V)+W = vector U+(v+W)
b) k(vector U+V) = k vector U + k vector V
c) (k+m)vector U = k vector U + m vector U
U+V + W = (u1+v1, u2+v2)+(w1,w2)
= (u1+v1+w1 , u2+v2+w2)
=(u1,u2) + (v1+w1`, v2+w2)
= U + (V+W)
Do the others the same way.
To prove each property using Cartesian vectors, we can break down the vectors and perform operations on individual components.
a) To prove (vector U+V)+W = vector U+(V+W), we will expand both sides of the equation and show that they are equal.
On the left-hand side:
(vector U+V)+W = (u1+v1, u2+v2) + (w1, w2) = (u1+v1+w1, u2+v2+w2)
On the right-hand side:
vector U+(V+W) = (u1, u2) + ((v1+w1, v2+w2)) = (u1+v1+w1, u2+v2+w2)
The left-hand side is equal to the right-hand side, thus proving the property.
b) To prove k(vector U+V) = k vector U + k vector V, we will apply the scalar multiplication to both sides of the equation.
On the left-hand side:
k(vector U+V) = k(u1+v1, u2+v2) = (k(u1+v1), k(u2+v2)) = (ku1+kv1, ku2+kv2)
On the right-hand side:
k vector U + k vector V = k(u1, u2) + k(v1, v2) = (ku1, ku2) + (kv1, kv2) = (ku1+kv1, ku2+kv2)
The left-hand side is equal to the right-hand side, thus proving the property.
c) To prove (k+m)vector U = k vector U + m vector U, we will distribute the scalar multiplication on the left-hand side and show that it is equal to the sum of scalar multiplications on the right-hand side.
On the left-hand side:
(k+m)vector U = (k+m)(u1, u2) = ((k+m)u1, (k+m)u2)
On the right-hand side:
k vector U + m vector U = k(u1, u2) + m(u1, u2) = (ku1, ku2) + (mu1, mu2) = (ku1+mu1, ku2+mu2)
The left-hand side is equal to the right-hand side, thus proving the property.
By expanding the vectors and applying the respective operations to the components, we have shown that each property holds true for Cartesian vectors.