The Resultant of Two Vectors and the Cross Product Calculation

Understanding vector operations is fundamental in various fields such as physics and engineering. One key operation is the calculation of the resultant of two vectors and their cross product. This article explains these concepts and provides a detailed example with vectors of magnitudes 3 units and 4 units that result in a resultant of 1 unit.

Resultant of Two Vectors

Given two vectors mathbf{A} and mathbf{B} with magnitudes 3 units and 4 units respectively, and a resultant vector mathbf{R} with a magnitude of 1 unit, we can determine the angle between them using the formula for the magnitude of the resultant vector:

The magnitude of the resultant vector is given by: [ mathbf{R} sqrt{mathbf{A}^2 mathbf{B}^2 - 2 mathbf{A} mathbf{B} cos theta} ] Substituting the known values: [ 1 sqrt{3^2 4^2 - 2 cdot 3 cdot 4 cdot cos theta} ] Squaring both sides: [ 1 9 16 - 24 cos theta ] Which simplifies to: [ 1 25 - 24 cos theta ] Hence: [ 24 cos theta 24 ] Solving for cos theta: [ cos theta 1 ] Given that cos theta -1, the angle theta between the vectors is 180^circ, indicating that the vectors are in opposite directions.

Now, we can determine the magnitude of their cross product using the formula:

[ mathbf{A} times mathbf{B} mathbf{A} mathbf{B} sin theta ]

Substituting the known values: [ mathbf{A} times mathbf{B} 3 cdot 4 cdot sin 180^circ ] Since sin 180^circ 0: [ mathbf{A} times mathbf{B} 0 ]

Therefore, the magnitude of the cross product of the two vectors is

Collinearity and Cross Product

It is important to note that the given vectors violate the triangle inequality, indicating that they are collinear. In such cases, the cross product of any pair of these vectors is a null vector. This is a direct consequence of the vectors being parallel yet in opposite directions.

Vectors and Resultant Magnitude Explained

The resultant vector of two vectors can be calculated using the formula:

The resultant vector magnitude formula is: [ R^2 A^2 B^2 - 2AB cos theta ] For given vectors mathbf{A} 3 text{units}, mathbf{B} 4 text{units} and a resultant vector mathbf{R} 1 text{unit}, we have: [ 1^2 3^2 4^2 - 2 cdot 3 cdot 4 cdot cos theta ] This simplifies to: [ 1 25 - 24 cos theta ] Hence: [ 24 cos theta 24 ] Solving for cos theta: [ cos theta -1 ] Therefore, theta 180^circ, meaning the vectors are opposite to each other.

If the angle between two vectors is 180^circ, the cross product is zero:

[ mathbf{A} times mathbf{B} mathbf{A} mathbf{B} sin 180^circ 0 ]

Conclusion

The vector operations discussed in this article illustrate how the resultant and cross product of vectors are interrelated and are essential concepts in various fields. Understanding these operations is crucial for solving problems in physics, engineering, and other scientific disciplines.