Thread: Vectors

1. AB is a diameter of a circle centered at the origin O, and P is any point on the circumference of the circle.
Using the position vectors of A, B and P, prove (using a scalar product) that AP is perpendicular to BP.

2. Use a vector method to prove that the diagonals of the square OABC cross at right angles.


I’m confused, actually how to do these?

Originally Posted by geton

1. AB is a diameter of a circle centered at the origin O, and P is any point on the circumference of the circle.
Using the position vectors of A, B and P, prove (using a scalar product) that AP is perpendicular to BP.

2. Use a vector method to prove that the diagonals of the square OABC cross at right angles.


I’m confused, actually how to do these?

Let denote the angle

1. The position vector of A is (-r,0)
The position vector of B is (r, 0)
The position vector of P is

2. Then

and



3. Now calculate



And therefore the legs of a right triangle are perpendicular. (Thales theorem)

To the second question:

The cordinates of the squares vertices are O(0, 0), A(s, 0), B(s, s), C(0, s) where s means the side length.

Calculate the vectors of the diagonals and prove that the scalar product of these two vectors is zero.

Originally Posted by earboth

Let denote the angle

1. The position vector of A is (-r,0)
The position vector of B is (r, 0)
The position vector of P is

2. Then

and



3. Now calculate



And therefore the legs of a right triangle are perpendicular. (Thales theorem)

To the second question:

The cordinates of the squares vertices are O(0, 0), A(s, 0), B(s, s), C(0, s) where s means the side length.

Calculate the vectors of the diagonals and prove that the scalar product of these two vectors is zero.

Thank you so much