Is It Possible for the Magnitude of the Resultant Vector to Be Less Than Any of the Given Vectors?

The magnitude of the resultant vector of two given vectors can indeed be less than the magnitude of any of these vectors, depending on the direction and relative magnitudes of the vectors involved. This concept is critical in various fields, including physics, engineering, and mathematics. Let's explore this interesting phenomenon.

Understanding Vector Addition and Subtraction

First, let's clarify the basics. Vector addition is a mathematical operation where two vectors are combined to form a resultant vector. If we have two vectors ( mathbf{V} ) and ( mathbf{W} ), their resultant vector ( mathbf{R} ) is given by ( mathbf{R} mathbf{V} mathbf{W} ). On the other hand, vector subtraction is the operation of finding the difference between two vectors, which can be represented as ( mathbf{V} - mathbf{W} ).

Example 1: Sum of Vectors in Opposite Directions

One simple example involves two vectors of equal magnitudes but in opposite directions. Let's say we have vectors ( mathbf{V} ) and ( -mathbf{V} ). The resultant vector is given by: [ mathbf{V} (-mathbf{V}) mathbf{0} ]

Notice that the magnitude of the resultant vector ( mathbf{0} ) is 0, which is less than the magnitude of ( mathbf{V} ) if ( mathbf{V} eq mathbf{0} ).

General Case: Resultant Vector Magnitude

In a more general case, let's consider vectors ( mathbf{A} ) and ( mathbf{B} ), where ( mathbf{B} ) is in the direction of and not necessarily equal to ( mathbf{A} ). The magnitude of the resultant vector ( mathbf{R} ) is influenced by the angle ( theta ) between the vectors. The resultant vector's magnitude ( R ) can be determined using the formula: [ R sqrt{A^2 B^2 2AB cos(theta)} ]

The range of this resultant magnitude ( R ) can vary from ( |B - A| ) to ( A B ), depending on the angle ( theta ). Specifically, if ( theta 90^circ ), ( R ) simplifies to ( sqrt{A^2 B^2} ), which can be less than either ( A ) or ( B ).

Special Case: Magnitude Difference and Angle

Let's further explore the special case where ( B > A ). According to the formula, the magnitude of the resultant vector will be less than ( B ) if ( B ) is less than twice the magnitude of ( A ). This is because, for angles ( theta ) exceeding ( 90^circ ), the cosine of the angle will be negative, thus reducing the overall magnitude of the resultant vector.

Example

Consider two vectors with magnitudes ( 2 ) and ( 2 ). If one vector points north and the other points 60 degrees southeast, the resultant vector's magnitude can be calculated as: [ text{Resultant magnitude} 2 cos(75^circ) 0.52 ]

Here, the resultant vector is approximately 0.52, which is obviously much smaller than the original vector magnitudes.

Equilibrium and Resultant Vector Magnitude

Another interesting scenario is when vectors form an equilibrium system. Take the case of a person standing on the ground. There are two forces at play: the person's weight pulling them downwards and the contact force from the ground pushing them upwards. When the person is in equilibrium, the resultant force is zero. [ text{Weight upward} text{Weight downward} 0 ]

This situation also illustrates how the resultant vector can be zero, despite the original vectors having non-zero magnitudes.

Conclusion

In conclusion, the magnitude of the resultant vector of two given vectors can indeed be less than any of the given vectors, depending on their relative magnitudes and the angle between them. This principle is fundamental in understanding vector operations and plays a crucial role in various real-world applications. Whether it's physics, engineering, or any field that deals with vector quantities, the concept of vector addition and subtraction is invaluable.