Vector A has a magnitude of 6 meters, and Vector B has a magnitude of 4 meters. What is the angle between them if their resultant is:
(a) minimum,
(b) maximum,
(c) 3 meters, and
(d) 8 meters?
Answer (2)
The angle between the vectors varies based on the resultant. For a minimum resultant of 2 meters the angle is 180 degrees, for a maximum of 10 meters it's 0 degrees. Specific angles for a 3-meter resultant is about 24.2 degrees and for an 8-meter resultant is about 104.5 degrees.
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To determine the angle between two vectors when their resultant has a specific magnitude, we use the law of cosines for vector addition. If Vector A has a magnitude A = 6 meters and Vector B has a magnitude B = 4 meters, and the angle between them is θ , the magnitude of the resultant vector R can be calculated as follows:
R 2 = A 2 + B 2 + 2 A B cos ( θ )
Let's solve the question in parts:
(a) Minimum Resultant :
The resultant is minimum when the two vectors are in opposite directions. In this case, the angle θ is 180 degrees (or π radians).
[
R = |A - B| = |6 - 4| = 2 \text{ meters} ]
(b) Maximum Resultant :
The resultant is maximum when the two vectors are in the same direction. In this case, the angle θ is 0 degrees.
[
R = A + B = 6 + 4 = 10 \text{ meters} ]
(c) Resultant of 3 meters :
To find the angle θ when the resultant is 3 meters:
[
3^2 = 6^2 + 4^2 + 2 \times 6 \times 4 \times \cos(\theta) ]
9 = 36 + 16 + 48 cos ( θ )
9 = 52 + 48 cos ( θ )
48 cos ( θ ) = 9 − 52 = − 43
cos ( θ ) = 48 − 43
Solve for θ using a calculator. This result implies the angle is greater than 90 degrees (since it leads to a negative cosine value), meaning they are not in the same direction.
(d) Resultant of 8 meters :
Set the resultant R = 8 meters:
[
8^2 = 6^2 + 4^2 + 2 \times 6 \times 4 \times \cos(\theta) ]
64 = 36 + 16 + 48 cos ( θ )
64 = 52 + 48 cos ( θ )
48 cos ( θ ) = 64 − 52 = 12
cos ( θ ) = 48 12 = 4 1
Use a calculator to find θ , which will be a value less than 90 degrees.
These calculations illustrate how vector magnitudes and the angles between them affect the resultant vector.
