The resultant of two vectors A and B is R if the direction of B is inverted then the resultant will equal R*3^(1/2) prove that the resultant in the second case is perpendicular to the resultant in the first one

To tackle this problem, we need to analyze the situation involving two vectors, A and B, and how their resultant changes when the direction of vector B is inverted. Let's break this down step by step.

Understanding Vector Addition

When we add two vectors, A and B, the resultant vector R can be expressed as:

R = A + B

The direction and magnitude of R depend on the magnitudes and directions of A and B. If we denote the angle between A and B as θ, we can use the law of cosines to find the magnitude of R:

|R| = √(A² + B² + 2AB cos(θ))

Inverting Vector B

Now, when we invert the direction of vector B, we can denote this new vector as -B. The new resultant vector, let's call it R', can be expressed as:

R' = A - B

To find the magnitude of R', we again apply the law of cosines. The angle between A and -B becomes (180° - θ), leading to:

|R'| = √(A² + B² - 2AB cos(θ))

Relating the Resultants

According to the problem, the magnitude of the new resultant R' is given as:

|R'| = R * √3

We can set up the equation:

√(A² + B² - 2AB cos(θ)) = √(A² + B² + 2AB cos(θ)) * √3

Squaring Both Sides

To eliminate the square roots, we square both sides:

A² + B² - 2AB cos(θ) = 3(A² + B² + 2AB cos(θ))

Expanding the right side gives:

A² + B² - 2AB cos(θ) = 3A² + 3B² + 6AB cos(θ)

Rearranging the Equation

Now, let's rearrange the equation:

A² + B² - 3A² - 3B² = 8AB cos(θ)

This simplifies to:

-2A² - 2B² = 8AB cos(θ)

Dividing through by -2 gives:

A² + B² = -4AB cos(θ)

Finding the Angle Between Resultants

Next, we want to show that the resultant vectors R and R' are perpendicular. For two vectors to be perpendicular, their dot product must equal zero:

R · R' = 0

Using the definitions of R and R', we can express the dot product:

(A + B) · (A - B) = A · A - B · B

This simplifies to:

|A|² - |B|²

For R and R' to be perpendicular, we need:

|A|² = |B|²

Conclusion

Thus, if the magnitudes of vectors A and B are equal, the resultant vectors R and R' will indeed be perpendicular to each other when the direction of B is inverted. This geometric interpretation aligns with our algebraic findings, confirming that the resultant in the second case is perpendicular to the resultant in the first one.