Please determine the specifications of this vector a = (i,j,k) that makes an angle of 70 degrees with the y-axis, 20 degrees with the z axis and is perpindicular (normal) to the x-axis.  

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mlehuzzah eNotes educator | Certified Educator

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First, the question is a little ambiguous.

Picture the following in 2 dimensions:

<2,1>, <-2,1>, <-2,-1>, <2,-1>

All of these, in some sense, make the same angle to the x axis, and all make the same angle to the y-axis.

This is because the vectors <1,0> and <-1,0> both lie on the x-axis, but they point in opposite directions. The angle that <2,1> makes with <1,0> is different from the angle that <2,1> makes with <-1,0>.

So instead, let's make the question more precise by saying we want a vector that is 70 degrees from <0,1,0> (the positive y-axis), 20 degrees from <0,01> (the positive z-axis), and 90 degrees from <1,0,0>

Let's label the components, so the vector we want is: <a,b,c>

The vector is normal to the x axis. So:

`<a,b,c>*<1,0,0> =0`  (that . is the dot product)

So a*1 + b*0 + c*0 = 0, so a=0

So our vector is: <0,b,c>

Now, `b!=0` ` ` If the vector had b=0, then it would be parallel to the z-axis. This is not the case, so b is...

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