What is v×w if v, a, and b have length 1 and w has length 2?

[Lord Popo]

Homework Statement

Let v, a, and b be the vectors (in the plane of the paper), all of which have length 1. (See attached picture). Let w be a vector of length 2 pointing directly out of the paper. Which of the following vectors is v×w?

Homework Equations

The formula for the cross product of vectors v and w = v×w = (v₂w₃ - (v₃w₂)i - (v₁w₃ - (v₃w₁)j + (v₁w₂ - (v₂w₁)k

The formula for the length of the cross product is |v×w| = |v| |w| sin θ

The Attempt at a Solution

I'm not sure if I approached this correctly, but I assumed that the lengths of the vectors in the picture are: |v| = 1, |a| = 1, |b| = 1, and |w| = 2. Since w comes out of the paper, I'm assuming it's perpendicular to v, so the angle between v| and v| is 90°. Therefore, sin (90°) = 1
Then I found the length of the cross product of v and w = |v×w| = (1) (2) sin (90°) = (1)(2)(1) = 2.
I still don't understand why the answer to the problem is 2b, and how would we know exactly what all the vector components of v, a, and b would be? I've been stuck on this for a while. Thank you in advance for your help.

 

[RPinPA]

Then I found the length of the cross product

Right. That's not the cross product (which is a vector). That's its magnitude. To find its direction you can use the right hand rule, which you can imagine using either the fingers of your right hand, or a normal right-handed screw (which goes in when turned clockwise and out when turned counterclockwise).

v x w is a vector which is perpendicular to both v and w. Because w is directly out of the paper, that's going to be in the plane of the paper (everything in the plane of the paper is perpendicular to w). And it's perpendicular to v. You'll notice that b is perpendicular to v in the plane of the paper. So the cross product points either in the direction of b or the opposite direction, and applying the right-hand rule...

<< Mentor Note -- the end of this reply has been deleted due to a bit too much explicit equation help >>

 

[Orodruin]

You do not need to know what the components are. In fact, coordinate systems would just get in the way here. Instead, focus on the invariant geometrical properties of the vectors. For example, you have concluded (correctly) that the length of vxw is 2. What about the direction?

 

[Lord Popo]

Right. That's not the cross product (which is a vector). That's its magnitude. To find its direction you can use the right hand rule, which you can imagine using either the fingers of your right hand, or a normal right-handed screw (which goes in when turned clockwise and out when turned counterclockwise).

v x w is a vector which is perpendicular to both v and w. Because w is directly out of the paper, that's going to be in the plane of the paper (everything in the plane of the paper is perpendicular to w). And it's perpendicular to v. You'll notice that b is perpendicular to v in the plane of the paper. So the cross product points either in the direction of b or the opposite direction, and applying the right-hand rule...

<< Mentor Note -- the end of this reply has been deleted due to a bit too much explicit equation help >>

Hi, thank you for your response. I was wondering how do I know that v and b are perpendicular if the problem doesn't specify that they are? What if the figure was not drawn to scale?

 

[Orodruin]

Generally you don’t. It should be appreciated from the figure in this case. A box to mark the right angle would have helped the figure ...

 

[RPinPA]

They didn't really indicate that v is perpendicular to b and they should have, but the artificial situation of having a multiple-choice answer gives you that clue. It certainly isn't perpendicular to a.

And if it was so badly drawn that the vector they intended as perpendicular didn't look like that at all, then you'd at least have a right to complain about a bad question.