Determine the coordinates of a unit vector that is perpendicular to both a = and b = and makes an obtuse angle with the z-axis.

Question

Determine the coordinates of a unit vector that is perpendicular to both a = <1, 1, 1> and b = <1, 3 -1> and makes an obtuse angle with the z-axis.

anonymous · Answer

I calculated the cross-product- it's <-4, 2, 2> and calculated the unit vector of that, but the answer is wrong. I think it's because of the obtuse thing- is there a specific formula I should be using?

anonymous · Answer

We can represent the $z$ axis with the vector $\langle 0,0,1\rangle$.

anonymous · Answer

What I mean is, such is a vector that is parallel to the positive $z$ axis.

anonymous · Answer

So I find a vector parallel to the z-axis, but also perpendicular to a and b?

anonymous · Answer

No

anonymous · Answer

The reason I did this is because you can find the angle between to vectors.

anonymous · Answer

two vectors

anonymous · Answer

$$
|a||b|\cos\theta = |a\cdot b|
$$

anonymous · Answer

I think the question is a bit silly though, because it will always form an acute angle with the $z$ axis if you count the positive and negative... so I assume they're only considering the positive part of the axis.

anonymous · Answer

|dw:1418964908593:dw|
In general if we project onto the $xy$ plane we can see that our angle will be a right angle.

anonymous · Answer

Any vector with a positive $z$ value will be acute to the positive $z$ axis, and any vector with a negative $z$ value will be obtuse to the positive $z$ axis.

anonymous · Answer

Also... I should have put:$$
|a||b|\cos\theta = a\cdot b
$$

anonymous · Answer

Had we let $b=\langle 0,0,1\rangle$, then we would have $|b| = 1$ and $a\cdot b = a_z$

anonymous · Answer

I didn't realize that, ha ha. Um, so I'm sorry, but I'm still stuck- are there mutiple vectors that are perpendicular to two vectors? Because the only one I can find is <-4, 2, 2>, and that does not result in an obtuse angle with the positive z-axis.

anonymous · Answer

Consider that if $a\cdot b = 0$, then $(-a)\cdot b = -a\cdot b = 0$.

anonymous · Answer

We can always say that in $\mathbb R^3$ there will be two vectors orthogonal to any pair of vectors.

anonymous · Answer

$$
u\times v = -(v\times u)
$$

anonymous · Answer

So, if you did the cross of $a$ and $b$, maybe try $b$ and $a$.  It's not a commutative operation.

anonymous · Answer

Alright, thanks, this one is obtuse. Unit vector calculation is: $$unit vector =\frac{ v }{ |v| }$$ right? just making sure.

anonymous · Answer

Alright, got the answer! Thanks for your help!