Problem 11 - Advanced Calculus

Question

Draw a diagram to illustrate that the associative law of addition, \(p+(q+r)=(p+q)+r\), holds for the operation of addition of vectors.

Step-by-Step Solution

Answer

It can be observed from the drawn diagram that the resultant vectors for both \(p+(q+r)\) and \((p+q)+r\) coincide. This visually demonstrates that the associative law of addition holds for vector addition.

Step 1: Understand the Associative Law of Addition

The associative law of addition states that when three or more numbers are added, the sum is the same regardless of the grouping of the addends. This can be represented mathematically as \(p+(q+r)=(p+q)+r\). This law is true for vectors as well. It means that for any vectors p, q and r the grouping in which they are added does not affect the final result.

Step 2: Draw the vectors

We begin by illustrating vectors p, q, and r in a diagram. Let’s assume that each vector has its starting point at origin (0,0).

Step 3: Illustrated \(p+(q+r)\)

Starting from the origin, draw the vector p, then at the head of vector p, draw the vector q. Lastly, at the head of vector q, draw the vector r. The resultant vector \(p+(q+r)\) will be the vector from origin directly to the head of vector r.

Step 4: Illustrate \((p+q)+r\)

Starting from the origin, draw the vector p, then at the head of vector p, draw the vector q. This forms the vector for \(p+q\). Now, at the head of vector \(p+q\), draw the vector r. The resultant vector \((p+q)+r\) will be the vector from origin directly to the head of vector r.

Step 5: The final conclusion of the diagram

If the two vectors \(p+(q+r)\) and \((p+q)+r\) coincide, this suggests that the associative law holds for vector addition, since the diagrams for \(p+(q+r)\) and \((p+q)+r\) provide the same resultant.

Key Concepts

Associative Law

The Associative Law is an essential property of addition in mathematics. It applies not just to numbers, but to vectors as well, which are mathematical entities that have both direction and magnitude. According to the associative property of addition, the way in which addends are grouped does not change the resultant sum. In the context of vectors, it states:

• For vectors p, q, and r, the equation \(p+(q+r) = (p+q)+r\) holds true.
• This property confirms that the grouping of vectors when added does not influence the final vector sum.

Thus, whether you first add vector q to vector r and then add the result to vector p, or first combine vector p and vector q and then add vector r, the final outcome remains the same.

Vector Diagrams

Vector diagrams are a great tool for visually representing vector addition and understanding concepts like the Associative Law. A vector is typically a directed line segment in these diagrams:

• The length of the segment represents the vector's magnitude.
• The direction in which the segment points reflects the vector's direction.

To illustrate the associative property with a vector diagram, you can draw:

• Starting at the origin, draw vector p.
• From the tip of p, draw vector q, and then from the tip of q, draw vector r. The end of vector r represents \(p+(q+r)\).
• Separately, draw vector p followed by vector q again to form p+q, then add vector r at the tip of p+q to observe the vector \((p+q)+r\).

Both methods should yield a resultant vector pointing to the same endpoint, evidencing the associative property of vector addition.

Mathematical Proofs

Mathematical proofs are formal ways to demonstrate the truth of a mathematical statement or property, such as the associative property of vectors. While visual proofs using diagrams can provide an intuitive understanding, formal proofs involve calculations and logical reasoning:

• Consider vectors represented in component form as \( \textbf{p}=(p_x, p_y) \), \( \textbf{q}=(q_x, q_y) \), and \( \textbf{r}=(r_x, r_y) \).
• Applying the associative property, we calculate \( \textbf{p} + (\textbf{q} + \textbf{r}) = (p_x + (q_x + r_x), p_y + (q_y + r_y)) \).
• This should be equal to \( (\textbf{p} + \textbf{q}) + \textbf{r} = ((p_x + q_x) + r_x, (p_y + q_y) + r_y) \).
• The results of both expressions are identical, confirming that the associative law holds algebraically as it does visually in diagrams.

Understanding both the visual and algebraic proofs of vector addition can lead to a deeper comprehension of vector properties and facilitate solving complex problems involving vectors.