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You MUST use vectors operations and/or dot product to solve this question. Any other...
You MUST use vectors operations and/or dot product to solve this question. Any other method will not be accepted.
Let A and B be the endpoints of a diameter of a circle. Let C be any point of the same circle. Show that the segments CA and CB are perpendicular.
Hint: I recommend using vector v =vector AM and vector w = vector MC
1 Answer | add yours
(Level 1) Associate Educator, Expert, Newton
The middle is M; the tip of the red arrow is A; the tip of the blue arrow is B; the tip of all the other arrows is C
red arrow: `vec(MA)=-vec(v)`
blue arrow: `vec(MB)=vec(v)`
green arrow: `vec(MC)=vec(w)`
purple arrow: `vec(BC)`
orange arrow: `vec(AC)`
green = red+orange, so:
orange = green - red
purple = (-blue) + green
orange = green-red = `vec(w)-(-vec(v))`
purple = (-blue)+green = `-vec(v)+vec(w)`
To show that the orange and purple vectors are perpendicular, we take the dot product and show it is zero:
`(vec(w)+vec(v)) * (vec(w)-vec(v))=`
`||vec(w)||^2 - vec(v)*vec(w) + vec(v)*vec(w) - ||vec(v)||^2 =`
`||vec(w)||^2 - ||vec(v)||^2 `
but these are the same length because both `vec(v)` and `vec(w)` are radii, and so have the same length
`||vec(w)||^2 - ||vec(v)||^2 =0`
So the vectors are perpendicular
Posted by mlehuzzah on January 22, 2013 at 2:21 AM (Answer #1)
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