Question

Show on a diagram three arbitrary vectors \(\boldsymbol{p}, \boldsymbol{q}\) and \(\boldsymbol{r}\). Using the triangle law of addition verify the associativity rule $$ p+(q+\boldsymbol{r})=(p+q)+\boldsymbol{r} $$

Short answer

Question: Prove the associativity rule in vector addition using the triangle law of addition. Answer: To prove the associativity rule in vector addition, we first draw three arbitrary vectors, add \(\boldsymbol{q}\) and \(\boldsymbol{r}\) using the triangle law, and then add \(\boldsymbol{p}\) to the sum. This is represented by the vector \(p+(q+\boldsymbol{r})\). Next, we add \(\boldsymbol{p}\) and \(\boldsymbol{q}\) using the triangle law and then add the sum to \(\boldsymbol{r}\), which is represented by \((p+q)+\boldsymbol{r}\). By observing that both of these vectors arrive at the same point and represent the same vector, we prove that the associativity rule is true: $$ p+(q+\boldsymbol{r})=(p+q)+\boldsymbol{r} $$

Step-by-step solution

Draw a diagram of three arbitrary vectors

Draw three arbitrary vectors, \(\boldsymbol{p}, \boldsymbol{q},\) and \(\boldsymbol{r}\) with their tails at the origin of a coordinate system. Label these as \(\boldsymbol{p}, \boldsymbol{q},\) and \(\boldsymbol{r}\) respectively.

Add \(\boldsymbol{q}\) and \(\boldsymbol{r}\) using the triangle law

To add \(\boldsymbol{q}\) and \(\boldsymbol{r}\), move the tail of \(\boldsymbol{r}\) to the head of \(\boldsymbol{q}\). The sum of \(\boldsymbol{q}\) and \(\boldsymbol{r}\) is represented by a vector that goes from the tail of \(\boldsymbol{q}\) to the head of \(\boldsymbol{r}\), denoted by \(\boldsymbol{q} + \boldsymbol{r}\). Label this new vector.

Add \(\boldsymbol{p}\) to the sum of \(\boldsymbol{q}\) and \(\boldsymbol{r}\)

To add \(\boldsymbol{p}\) to the sum of \(\boldsymbol{q}\) and \(\boldsymbol{r}\), move the tail of \(\boldsymbol{p}\) to the head of \(\boldsymbol{q} + \boldsymbol{r}\). The sum of \(\boldsymbol{p},\) \(\boldsymbol{q},\) and \(\boldsymbol{r}\) is represented by a vector that goes from the tail of \(\boldsymbol{p}\) to the head of \(\boldsymbol{q} + \boldsymbol{r}\). Label this new vector as \(p+(q+\boldsymbol{r})\).

Add \(\boldsymbol{p}\) and \(\boldsymbol{q}\) using the triangle law

To add \(\boldsymbol{p}\) and \(\boldsymbol{q}\), move the tail of \(\boldsymbol{q}\) to the head of \(\boldsymbol{p}\). The sum of \(\boldsymbol{p}\) and \(\boldsymbol{q}\) is represented by a vector that goes from the tail of \(\boldsymbol{p}\) to the head of \(\boldsymbol{q}\), denoted by \(\boldsymbol{p} + \boldsymbol{q}\). Label this new vector.

Add the sum of \(\boldsymbol{p}\) and \(\boldsymbol{q}\) to \(\boldsymbol{r}\)

To add the sum of \(\boldsymbol{p}\) and \(\boldsymbol{q}\) to \(\boldsymbol{r}\), move the tail of \(\boldsymbol{r}\) to the head of \(\boldsymbol{p} + \boldsymbol{q}\). The sum of \(\boldsymbol{p},\) \(\boldsymbol{q},\) and \(\boldsymbol{r}\) is represented by a vector that goes from the tail of \(\boldsymbol{p} + \boldsymbol{q}\) to the head of \(\boldsymbol{r}\). Label this new vector as \((p+q)+\boldsymbol{r}\).

Verify the associativity rule

Observe the diagrams in steps 3 and 5. Notice that both \(p+(q+\boldsymbol{r})\) and \((p+q)+\boldsymbol{r}\) represent the same vector and arrive at the same point when starting from the origin. Therefore, the associativity rule holds true, as $$ p+(q+\boldsymbol{r})=(p+q)+\boldsymbol{r} $$

Key concepts

Vector Associativity

When dealing with vectors, understanding the principle of vector associativity is essential for anyone studying mathematics, especially in fields such as physics and engineering. In essence, this property allows us to group vectors together in a sum without affecting the result, granting a great deal of flexibility in solving problems involving multiple vectors.

By definition, vector associativity signifies that when you are adding more than two vectors, the way in which they are grouped does not change the final vector sum. Mathematically, it is expressed as \(\vec{a} + (\vec{b} + \vec{c}) = (\vec{a} + \vec{b}) + \vec{c}\), with \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) representing any vectors in space.

For example, if you’re working on an engineering project and need to find the resultant force acting on a body, the forces can be represented as vectors. It doesn't matter in which order you sum up the individual force vectors; the resultant vector, which combines all these forces, will be the same. This concept is also crucial when working with complex systems where vectors are added in different stages or parts. By simplifying calculations using associativity, students and engineers can streamline processes, minimize errors, and achieve more accurate results.

Vector Addition Diagram

A vector addition diagram is a visual representation that helps in the understanding of how vectors add together. This graphical tool plays a vital role in 'seeing' what the mathematical notation means in a practical sense and is particularly helpful for visual learners and those new to the study of vectors.

In a typical vector addition diagram, vectors are represented as arrows with a specific direction and magnitude. The triangle law of vector addition, which states that when two vectors are placed head to tail, the resultant vector is represented by the closing side of the triangle in the opposite direction, is a cornerstone of these diagrams. This visualization makes it clear why \(\vec{p} + (\vec{q} + \vec{r}) = (\vec{p} + \vec{q}) + \vec{r}\), as both ways lead to the same geometric representation.

The step-by-step solution provided in the exercise showcases the creation of such a diagram, demonstrating the practical application of vector addition and the verification of vector associativity. Students can benefit from practicing with vector addition diagrams to solidify their understanding and improve their ability to work with vectors in a range of contexts, from homework problems to real-world engineering challenges.

Mathematics for Engineers

The field of engineering is deeply intertwined with mathematics, as it provides the foundational tools required to analyze, model, and solve engineering problems. Knowledge of vector mathematics, including vector addition and the triangle law, is essential for engineers across various disciplines such as mechanical, civil, electrical, and aerospace engineering.

Engineers use vectors to represent quantities that have both magnitude and direction, such as forces, velocities, and accelerations. Understanding how to manipulate these vectors through addition is critical when analyzing structures, designing machinery, and programming simulations.

Associativity, as demonstrated in the exercise, is just one example of the many mathematical principles that are applied in engineering disciplines. The ability to visualize the combination of forces or other vector quantities on a vector addition diagram allows engineers to predict the behavior of complex systems and design solutions that are safe, efficient, and reliable. Thus, mastering vector mathematics is not merely an academic requirement, but a practical necessity for any aspiring or practicing engineer.