You have two vectors A and B. You are told that the component of A in the direction of B is zero. What does this tell you about the relative orientation of the two vectors? Explain your reasoning. Make a drawing to accompany your explanation.

Question

aum · Answer

A must be perpendicular to B.  Or, Acos(90) = 0

mathstudent55 · Answer

Here is vector B.
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mathstudent55 · Answer

Let's say I try a random vector A.

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aum · Answer

Yes.  If you assume the angle between A and B is t, then the component of A in the direction of B is A * cos(t).  It is given that the component is zero.
A * cos(t) = 0  implies cos(t) = 0  or t = 90 degrees.

mathstudent55 · Answer

Now we break up A into two components, parallel and perpendicular to B.

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mathstudent55 · Answer

Notice that for A to have no component in the direction of vector B, A must have only the component that is perpendicular to B. That means vector A must be perpendicular to vector B.

If you eliminate the component of A that is parallel to B, you are left with only the perpendicular component. That means A is perpendicular to B.

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