Two equal vectors are working at a point.The resultant is equal to one of those vectors.Prove that the resultant bisects the angle between the vectors.
Let's denote the two equal vectors as vector A and vector B, and the resultant vector as vector R.
Given that the magnitude of vector R is equal to one of the vectors (let's say vector A), we can represent this condition as |R| = |A|.
To prove that the resultant vector bisects the angle between the vectors, we need to show that the angle between vector R and vector A is equal to the angle between vector R and vector B.
We can use the dot product between two vectors to determine the angle between them. The dot product of two vectors can be calculated as follows:
A · B = |A| |B| cos(θ)
Where θ denotes the angle between vector A and vector B.
Since the magnitude of vector R is equal to the magnitude of vector A, we can rewrite the dot product equation as:
R · B = |R| |B| cos(θ)
Since |R| = |A|, we can substitute |A| for |R| in the equation:
R · B = |A| |B| cos(θ)
Given that R = A by assumption, the equation becomes:
A · B = |A| |B| cos(θ)
Since A and B are equal in magnitude, we can simplify the equation further:
A · B = A |B| cos(θ)
Dividing both sides by |A|:
B = |B| cos(θ)
By rearranging the equation, we get:
cos(θ) = 1
This equation implies that the angle θ between vector A and vector B is 0 degrees, which means they are collinear. In other words, vector A and vector B are pointing in the same direction.
Now, let's consider the angle between vector R and vector A. Since vector R equals vector A in magnitude, the angle between these two vectors is 0 degrees as well.
Therefore, the angle between vector R and vector A is equal to the angle between vector R and vector B, satisfying the condition that the resultant vector bisects the angle between the vectors.
To prove that the resultant bisects the angle between the two vectors, we need to apply vector addition and examine the properties of the resulting triangle.
Let's denote the two equal vectors as A and B. We are given that their resultant, R, is equal to one of the vectors. Now, let's analyze the problem step by step:
1. Given that the resultant R is equal to vector A, we can write it as R = A.
2. According to the problem statement, we have two equal vectors A and B. Therefore, we can write A = B.
3. Now, we can rewrite the equation R = A as R = B.
4. By comparing equations R = A and R = B, we can conclude that A = B.
Since A and B are equal vectors, the triangle formed by A, B, and R is an isosceles triangle.
5. In an isosceles triangle, the two equal sides are subtended by equal angles opposite to them.
6. In our case, A and B are the two equal sides of triangle ABR, and R is the angle between them.
7. Since A = B, the angles opposite to A and B are also equal. Therefore, the angle opposite to R is also equal to the angle opposite to A (or B).
8. Hence, we can conclude that the resultant R bisects the angle between the two vectors A and B.
In summary, by proving that the two equal vectors A and B are equivalent to the resultant R and that the resulting triangle is isosceles, we can deduce that the resultant bisects the angle between the vectors.