Resultant Vector Magnitude Range

When two vectors are added, their resultant vector's magnitude depends on the angle between them. The magnitude of the resultant vector can vary between a minimum value and a maximum value.

Calculating Minimum and Maximum Resultant Magnitudes

Let's consider two vectors, \(\vec{A}\) and \(\vec{B}\), with magnitudes \(A\) and \(B\) respectively.

• Minimum Resultant Magnitude: The minimum magnitude of the resultant vector occurs when the two vectors are acting in opposite directions (i.e., the angle between them is 180°). In this case, the magnitude of the resultant vector is the absolute difference of their magnitudes.
  Minimum Resultant \(= |A - B|\)
• Maximum Resultant Magnitude: The maximum magnitude of the resultant vector occurs when the two vectors are acting in the same direction (i.e., the angle between them is 0°). In this case, the magnitude of the resultant vector is the sum of their magnitudes.
  Maximum Resultant \(= A + B\)

For any other angle between 0° and 180°, the resultant magnitude will lie between these minimum and maximum values. Therefore, the possible range for the resultant magnitude \(R\) is:

\[|A - B| \le R \le A + B\]

Applying to the Given Vectors

We are given two vectors with magnitudes:

• Magnitude of first vector, \(A = 15\) units
• Magnitude of second vector, \(B = 10\) units

1. Calculating the Minimum Possible Resultant Magnitude:

When the vectors are anti-parallel (pointing in opposite directions):

\[R_{min} = |A - B| = |15 - 10| = |5| = 5 \text{ units}\]

2. Calculating the Maximum Possible Resultant Magnitude:

When the vectors are parallel (pointing in the same direction):

\[R_{max} = A + B = 15 + 10 = 25 \text{ units}\]

So, the magnitude of the resultant vector, \(R\), must always be within the range of 5 units to 25 units, inclusive. That is, \(5 \le R \le 25\).

Analyzing the Options

We need to find the option that falls outside this possible range \([5, 25]\).

From the analysis, a resultant magnitude of 3 units is less than the minimum possible resultant magnitude of 5 units. Therefore, the magnitude of the resultant vector of these two vectors can never be 3 units.