Understanding Vector Addition and Resultant Magnitude Calculation

When dealing with vectors, one fundamental question often arises: how do you add two vectors together, and what is the magnitude of their resultant?

Key Concepts in Vector Addition

Two vectors can be addition if they are of the same dimension and useful in the same context. However, if they are in opposite directions and their magnitudes cancel each other out, they cannot be added in a way that makes physical or mathematical sense. Let's explore how to calculate the resultant magnitude of vectors under different conditions.

Opposite Direction Vectors

If we have two vectors A with a magnitude of 3 and B with a magnitude of 4, pointing in opposite directions, their resultant vector's magnitude is the difference between their magnitudes. This is because their components in the opposite directions cancel out, leaving only the difference.

Thus, the magnitude of the resultant vector is:

Resultant Magnitude 4 - 3 1

Vectors in Different Directions

If the vectors are not in opposite directions, you can calculate their resultant magnitude using the concept of the dot product. Let A have a length of 3 and B have a length of 4. The expression for the dot product can be written as:

AB2 A2 B2 - 2A.B cos-angular between A and B

This simplifies to:

AB2 9 16 - 2(3)(4)cos∠AB

Because cos/angular runs between -1 and 1, AB2 ranges between 1 and 49. Therefore, the resultant length norm lies between 1 and 7. This means:

Resultant length is between 1 and 7

Three-Dimensional Vectors

If the vectors are in three dimensions, the process involves calculating the components in the x, y, and z directions. Algebraically, you add the x, y, and z components to get a resultant xyz vector Rxyz. The magnitude of the resultant vector can then be calculated using the Pythagorean theorem.

Example:

Axyz Bxyz with each component being 1. Rx 2, Ry 2, Rz 2 Restot √(22 22 22) √12 ≈ 3.464

If the vectors are not in the x and y planes, you would need the full 3D components to calculate the resultant. In the case where the vectors are in the x and y planes, the calculation simplifies to:

Resultant R √(42 32) √25 5

Conclusion

Understanding vector addition and resultant magnitude is essential for various fields, including physics, engineering, and mathematics. The key is to consider the direction and magnitude of the vectors involved, whether in two or three dimensions. Remember that the resultant length must be between the sum and the difference of the two vector lengths.

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