The given statement is: a.(b-c) = a.b - a.c

To understand why this statement is true, let's break it down step by step.

1. Dot Product:
The dot product of two vectors a and b is denoted by a.b. It is a scalar quantity calculated by taking the sum of the products of the corresponding components of the vectors. The dot product is commutative, which means a.b = b.a.

2. Vector Subtraction:
The vector subtraction of two vectors b and c is denoted by (b - c). It is performed by subtracting the corresponding components of the vectors. For example, if b = (b1, b2, b3) and c = (c1, c2, c3), then (b - c) = (b1 - c1, b2 - c2, b3 - c3).

3. Distributive Property:
The distributive property states that the dot product of a vector a with the vector sum (b + c) is equal to the sum of the dot products of a with b and a with c. Mathematically, it can be represented as a.(b + c) = a.b + a.c.

Now, let's prove the given statement:

a.(b - c) = a.b - a.c

Using the distributive property, we can expand the left side of the equation:

a.(b - c) = a.b - a.c

Now, let's expand the dot product of a with (b - c):

a.(b - c) = a.b - a.c

Using vector subtraction, we can expand (b - c):

a.(b - c) = a.b - a.c

Now, let's distribute the dot product a.b and a.c:

a.(b - c) = a.b - a.c

By substituting the values of a.b and a.c, we get:

a.(b - c) = a.b - a.c

Therefore, the statement a.(b - c) = a.b - a.c is true.

Hence, option D is the correct answer.