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vectors question

A, B and C have coordinates ( 1, 4, 2 ), ( 4, 6, 1 ) and ( 2, 5, 7 )

(i) Find the vectors AB and AC

(ii) Hence show that the vectors AB and AC are perpendicular

Im confused with part (ii), I've found AB=(3,2,-1) and AC=(1,1,5) and shown that AB.AC=0

What I then don't get is why you have to use 0=AC x AB x cosθ (checked the solutions), I don't understand how using that^ shows that the vectors are perpendicular but the dot product isn't enough to show they are perpendicular?
Reply 1
Avatar for joostan
joostan
13
Original post by mica-lwe
A, B and C have coordinates ( 1, 4, 2 ), ( 4, 6, 1 ) and ( 2, 5, 7 )

(i) Find the vectors AB and AC

(ii) Hence show that the vectors AB and AC are perpendicular

Im confused with part (ii), I've found AB=(3,2,-1) and AC=(1,1,5) and shown that AB.AC=0

What I then don't get is why you have to use 0=AC x AB x cosθ (checked the solutions), I don't understand how using that^ shows that the vectors are perpendicular but the dot product isn't enough to show they are perpendicular?

Since the magnitude of AC and AB is non-zero the dot product is zero must imply that cos⁡(θ)=0\cos(\theta)=0, which then means that the vectors must be orthogonal.
Reply 2
Avatar for mica-lwe
mica-lwe
OP
2
Thanks! :smile:
Reply 3
Avatar for joostan
joostan
13
Original post by mica-lwe
Thanks! :smile:


No worries.
This fact is quite useful and is worth remembering. :smile:

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