The resultant of two vectors of equal magnitude will lie exactly on the angle bisector between the two vectors.
Understanding Vector Resultants
When two vectors are added, their resultant is a single vector that represents the combined effect of the original vectors. The magnitude and direction of the resultant depend on the magnitudes and directions of the original vectors.
For two vectors, say A and B, the resultant R can be found using the parallelogram law of vector addition. If you place the tails of A and B together, they form two adjacent sides of a parallelogram. The diagonal of the parallelogram starting from the same point is the resultant R = A + B.
The Case of Equal Magnitudes
When the two vectors, A and B, have equal magnitudes (i.e., |A| = |B|), the parallelogram formed is a special type called a rhombus. A key property of a rhombus is that its diagonals bisect the angles at the vertices.
In this scenario:
- The two vectors A and B represent two adjacent sides of the rhombus.
- The resultant vector R is the diagonal of the rhombus originating from the point where A and B meet.
Since the diagonal of a rhombus bisects the angle, the resultant vector R will bisect the angle between vectors A and B.
Relationship to Magnitude Differences
The provided reference highlights what happens when vector magnitudes are not equal: "if ∣A ∣≠∣B ∣ then R will incline more towards the vector of bigger magnitude." This statement provides a contrasting scenario that reinforces the principle for equal magnitudes.
- Unequal Magnitudes: As stated in the reference, if one vector is larger than the other, the resultant will be closer in direction to the vector with the greater magnitude. It will not lie on the angle bisector.
- Equal Magnitudes: When |A| = |B|, there is no "bigger magnitude" vector. The resultant cannot incline more towards one vector than the other. Logically, this means the resultant must incline equally towards both, which is precisely the definition of lying on the angle bisector.
Therefore, the reference supports the understanding that the bisector is the location of the resultant specifically when the magnitudes are equal, by describing the deviation that occurs when they are unequal.
Summary
Here’s a quick summary of the direction of the resultant relative to the angle bisector:
| Vector Magnitudes | Direction of Resultant |
|---|---|
| Equal ( | A |
| Unequal ( | A |
Practical Application
This principle is useful in physics and engineering when dealing with forces, velocities, or other vector quantities. For instance:
- Forces: If two equal forces are applied to an object at an angle, the net force (resultant) acts along the bisector of the angle between the forces.
- Reflections: In optics, if incoming and outgoing light rays are represented as vectors of equal magnitude (speed), the change in direction vector is related to the bisector of the angle between the rays.
In conclusion, when two vectors possess identical magnitudes, their vector sum, or resultant, will precisely align with the line that bisects the angle separating the two vectors.
