How to Determine the Resultant Displacement of Given Vector Components

To determine the resultant displacement when given multiple vectors in terms of their components, you will combine the corresponding components of each vector and then calculate the magnitude of the resulting vector.

Displacement is a vector quantity, meaning it has both magnitude (how far) and direction. When you have multiple displacements happening sequentially or simultaneously, the resultant displacement is the single straight-line vector from the initial position to the final position.

Combining Vector Components

The first step in finding the resultant displacement vector from its components is to combine the components of all the individual displacement vectors.

As the reference states, to find the resultant displacement vector when given multiple vectors in unit vector notation, add the x, y, and z components of each vector together.

Let's say you have several displacement vectors:

• Vector A = Ax i + Ay j + Az k
• Vector B = Bx i + By j + Bz k
• ...and so on for any number of vectors.

The resultant displacement vector, R, will have components (Rx, Ry, Rz) found by summing the corresponding components:

• Rx = Ax + Bx + ... (Sum of all x-components)
• Ry = Ay + By + ... (Sum of all y-components)
• Rz = Az + Bz + ... (Sum of all z-components)

The resultant vector can then be written as R = Rx i + Ry j + Rz k. This resultant vector R represents the total displacement, including its direction.

Calculating the Magnitude of Resultant Displacement

Often, you need to know the total distance covered in a straight line from start to finish, which is the magnitude of the resultant displacement vector.

The magnitude of the resultant vector R is calculated using the three-dimensional version of the Pythagorean theorem. The reference explicitly provides the formula:

R = √(Rx² + Ry² + Rz²)

Where:

• R is the magnitude of the resultant displacement.
• Rx, Ry, and Rz are the resultant components calculated in the previous step.

This calculation gives you a single numerical value representing the straight-line distance of the total displacement.

Practical Example

Let's find the resultant displacement and its magnitude for two displacements:

• Displacement d₁ = 3i + 4j meters
• Displacement d₂ = 2i - 2j + 1k meters

Here, we assume a z-component of 0 for d₁ since it's not explicitly given.

1. Combine Components:

   • Rx = (x-component of d₁) + (x-component of d₂) = 3 + 2 = 5 meters
   • Ry = (y-component of d₁) + (y-component of d₂) = 4 + (-2) = 2 meters
   • Rz = (z-component of d₁) + (z-component of d₂) = 0 + 1 = 1 meter

   The resultant displacement vector is R = 5i + 2j + 1k meters.

2. Calculate Magnitude:
   Using the formula R = √(Rx² + Ry² + Rz²):

   • R = √(5² + 2² + 1²)
   • R = √(25 + 4 + 1)
   • R = √30 meters

So, the resultant displacement vector is 5i + 2j + 1k meters, and its magnitude (the straight-line distance from start to finish) is √30 meters.

In summary, determining the resultant displacement from given vector components involves summing the corresponding components to find the resultant vector and then using the Pythagorean theorem formula provided to find the magnitude of that vector.