Prove that an angle inscribed in a semi-circle is a right angle using vector method.

Explanation

Let O be the centre of the semi-circle and BA be the diamter. Let P be any point on the cirumference of the semi-circle.
Let `vecOA=veca, "then"vecOB=-veca`
` Let " " vecOP=vecr`
`vecAP=vecOP-vecOA=vecr-veca`
`vecBP=vecOP-vecOB=vecr-(-veca)=vecr+veca`

`vecAP.vecBP=(vecr-veca).(vecr+veca)`
`= vec(r^(2))-vec(a^(2))`
`=a^(2)-a^(2)`
Therefore, `vec(AP)` is perpendicular to `vec(BP)` , i.e.
` Rightarrow angleAPB = 90^(@)`