Vectors Thinking Question: Proving Perpendicularity using the Cosine Law

[DespicableMe]

Homework Statement

a) Write the cosine law using a vector operation.
b) If |vector a - vector b| = |vector a + vector b|, prove that a is perpendicular to b using the formula you found in a).

The Attempt at a Solution

The red period represents the dot product

a) magnitude c = SQUAREROOT (a.a + b.b - 2(a.b) )

b) magnitude c = SQUAREROOT (a.a + b.b - 2(a.b) )

= SQUAREROOT ( a² + b² - 2( |a||b|cos 90 )
= SQUAREROOT ( a² + b² )

And this is the Pythagorean theorem, which only applies for right angle triangles, so if mag. C is SQUARERT (a²+b²), then a and b are perpendicular.

DId I do this correctly?

I didn't know how to include |vector a - vector b| = |vector a + vector b| into the equation and it only asked to prove it using the formula that I found...

 

[Mark44]

Homework Statement

a) Write the cosine law using a vector operation.
b) If |vector a - vector b| = |vector a + vector b|, prove that a is perpendicular to b using the formula you found in a).

The Attempt at a Solution

The red period represents the dot product

a) magnitude c = SQUAREROOT (a.a + b.b - 2(a.b) )

b) magnitude c = SQUAREROOT (a.a + b.b - 2(a.b) )

= SQUAREROOT ( a² + b² - 2( |a||b|cos 90 )
= SQUAREROOT ( a² + b² )

And this is the Pythagorean theorem, which only applies for right angle triangles, so if mag. C is SQUARERT (a²+b²), then a and b are perpendicular.

DId I do this correctly?

I didn't know how to include |vector a - vector b| = |vector a + vector b| into the equation and it only asked to prove it using the formula that I found...

The b part isn't correct. You need to start by assuming that |a - b| = |a + b|, and showing that it follows that a and b are perpendicular.

You could start by assigning coordinates to a and b, such as a = <a₁, a₂> and b = <b₁, b₂>.

 

[DespicableMe]

The b part isn't correct. You need to start by assuming that |a - b| = |a + b|, and showing that it follows that a and b are perpendicular.

You could start by assigning coordinates to a and b, such as a = <a₁, a₂> and b = <b₁, b₂>.

I'm still kind of...
What can I do with the coordinates?

At one point, I did draw vector diagrams to show|a - b| = |a + b| but other than that, I didn't know how to use that in the equation.

 

[Char. Limit]

Start by writing out the magnitudes using the definition of magnitude. Of course, it would be easier if you squared both sides first.