How to Find the Sum of Three Vectors?

Finding the sum of three vectors can be done systematically by adding them in sequence.

The most straightforward way to find the sum of three vectors is to add the first two vectors together to get a resultant vector, and then add this resultant vector to the third vector.

Step-by-Step Vector Addition

To illustrate this process, let's consider three vectors: C, D, and E. The goal is to find their sum, which we can call R'.

The process involves these key steps:

1. Sum the first two vectors: First, you add the vectors C and D together. This operation yields a new vector, often called the resultant vector of the first two, let's name it R.

   • Mathematically, this step is: R = C + D.

2. Sum the resultant with the third vector: Next, you take the resultant vector R obtained from the first step and add it to the third vector, E. The outcome of this addition is the final resultant vector, R', which represents the sum of all three original vectors.

   • Mathematically, this step is: R' = R + E.

Therefore, the overall sum is R' = (C + D) + E. This method is based directly on the principle mentioned in the reference: "For the addition of the three vectors, we can make the vector addition for the first two vectors C and D, where we obtain the resultant R. Then add this result to the third vector, E to obtain the final resultant of the addition of three vectors, i.e., R'."

Methods for Adding Vectors

Vectors can be added using graphical methods (like the parallelogram or head-to-tail method) or analytical methods (using components). For three vectors, the analytical method using components is usually the most efficient and accurate.

Analytical Method (Using Components)

If the vectors are given in component form (e.g., in a 2D or 3D coordinate system), you can add them by adding their corresponding components.

Let's assume the three vectors are:

• C = (c₁, c₂, c₃)
• D = (d₁, d₂, d₃)
• E = (e₁, e₂, e₃)

Using the step-by-step process:

1. Find the resultant of C + D:
   R = C + D = (c₁ + d₁, c₂ + d₂, c₃ + d₃)

2. Find the final resultant of R + E:
   R' = R + E = ((c₁ + d₁) + e₁, (c₂ + d₂) + e₂, (c₃ + d₃) + e₃)
   R' = (c₁ + d₁ + e₁, c₂ + d₂ + e₂, c₃ + d₃ + e₃)

This shows that the sum of three vectors is simply the sum of their corresponding components. The intermediate step of finding R is consistent with the reference and simplifies the conceptual understanding.

Example Using Components

Let's find the sum of three 2D vectors:

• C = (2, 3)
• D = (-1, 5)
• E = (4, -2)

1. Sum C + D:
   R = C + D = (2 + (-1), 3 + 5) = (1, 8)

2. Sum R + E:
   R' = R + E = (1 + 4, 8 + (-2)) = (5, 6)

So, the sum of the three vectors is R' = (5, 6).

Summary Table

Here's a brief summary of the process using component addition:

This method can be extended to any number of vectors and any dimension.

For more detailed information on vector addition principles, you can refer to resources on vector algebra (Learn More about Vector Addition - Placeholder Link).

In conclusion, summing three vectors is a process of sequential addition, typically best handled by summing their corresponding components.