If $A = <3 ,2 ,5 >$ , $B = <5 ,4 ,8 >$ and $C=A-B$ , what is the angle between A and C?

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Answer 1 · 2017-05-13T18:05:34

The angle is $=169.61$ º

Let's start by calculating

$vecC=vecA-vecB$

$vecC=〈3,2,5〉-〈5,4,8〉=〈-2,-2,-3〉$

The angle between $vecA$ and $vecC$ is given by the dot product definition.

$vecA.vecC=∥vecA∥*∥vecC∥costheta$

Where $theta$ is the angle between $vecA$ and $vecC$

The dot product is

$vecA.vecC=〈3,2,5〉.〈-2,-2,-3〉=-6-4-15=-25$

The modulus of $vecA$ = $∥〈3,2,5〉∥=sqrt(9+4+25)=sqrt38$

The modulus of $vecC$ = $∥〈-2,-2,-3〉∥=sqrt(4+4+9)=sqrt17$

So,

$costheta=(vecA.vecC)/(∥vecA∥*∥vecC∥)=-25/(sqrt38*sqrt17)=-0.984$

$theta=169.61$ º

If A = <3 ,2 ,5 >A = <3 ,2 ,5 >, B = <5 ,4 ,8 >B = <5 ,4 ,8 > and C=A-BC=A-B, what is the angle between A and C?