The vectors \(\mathbf{A}, \mathbf{B}, \mathbf{C}\) are coplanar. Show graphically that $$ \mathbf{A} \cdot(\mathbf{B}+\mathbf{C})=\mathbf{A} \cdot \mathbf{B}+\mathbf{A} \cdot \mathbf{C} $$

The distributive property is a mathematical principle that applies to many operations, including the vector dot product. When it comes to vectors, this property allows us to distribute the dot product over a sum of vectors. In simpler terms, if you have a vector \textbf{A} and are taking the dot product with the sum of two other vectors, \textbf{B} and \textbf{C}, you can compute the dot product of each pair individually and then sum the results. It's essentially a way to break down a more complex operation into simpler steps.

Mathematically, this is expressed as \( \mathbf{A} \cdot (\mathbf{B} + \mathbf{C}) = \mathbf{A} \cdot \mathbf{B} + \mathbf{A} \cdot \mathbf{C} \). The property holds for any number of vectors added together, and it is extremely useful for simplifying calculations in physics and engineering, where vectors are often combined.

Adding two vectors together is a fundamental operation in vector algebra. To add vectors, you simply align them head to tail and draw a new vector from the free tail to the free head. This new vector is called the resultant vector, and it effectively combines the magnitude and direction of the original vectors. This method is known as the tip-to-tail method.

The graphical method of vector addition not only helps in visualizing the vectors and their resultant but also in applying other vector operations like the dot product. It's very important to understand that vector addition is defined geometrically, and it is different from adding scalars because it takes into account both magnitude and direction.

Unlike vectors, which have both magnitude and direction, scalar quantities possess only magnitude. Examples of scalar quantities include temperature, mass, and time. In the context of vector multiplication, the dot product of two vectors yields a scalar quantity, not a vector. This scalar represents the product of the magnitudes of the two vectors and the cosine of the angle between them.

The scalar outcome of a dot product can be very telling; it can indicate whether two vectors are orthogonal (if the product is zero), how much one vector extends in the direction of another, or the projection of one vector onto another. Understanding the difference between vectors and scalars is crucial for grasping the various ways in which they interact in mathematical operations.

Illustrating vectors graphically allows for an intuitive understanding of vector operations. In a graphical representation, vectors are usually depicted as directed line segments. The direction of the arrow shows the direction of the vector, while its length represents the vector's magnitude, often drawn to a specific scale.

The visual approach is beneficial when learning about vector addition and the distributive property because it provides a clear picture of how vectors combine and interact. This can be especially helpful when dealing with the vector dot product, as projections can be visually analyzed and compared. A well-drawn vector diagram can serve as a powerful tool for both explanation and confirmation of vector operations.