(I) Show that (b) î x ĵ, = k̂ , î x k̂ , = - ĵ , and...

Question

(I) Show that (b) î x ĵ, = k̂ , î x k̂ , = - ĵ , and ĵ x k̂ = î .

Accepted Answer

Hello, fellow physicists today, we're solve the following practice problem together. So first off, let's read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Given I hat, J hat and K hat represent unit vectors along the positive X positive Y and positive Z axis respectfully determine the cross products of the following J hat and I hat, K hat and I hat and K hat and J hat. So our end goal is to find the cross product of J hat and I hat and K hat and I hat and K hat and J hat. So those are the three cross products that we're trying to solve for. So we're trying to figure out what they are equal to Awesome. We're also given some multiple choice answers. Let's read them off to see what our final answer pair might be or I should say answer set. So for a, it says that the cross products of J hat and I hat is equal to negative K hat and the cross products of K hat and I hat is equal to negative J hat and the cross products of khat in J hat is equal to I hat. And for B it says that the cross products of I hat, I mean J hat and I hat is equal to negative K hat. And the pro cross product of K hat and I hat is negative J hat and the cross product of K hat and J hat is equal to negative I hat. And for c it states that the cross product of J hat and I hat is equal to negative K hat and the cross product of K hat and I hat is equal to J hat. And the cross product of K hat and J hat is equal to negative I hat. And then finally, for D, the cross product of J hat and I hat is equal to K hat in the cross product of K hat and I hat is J hat. And the cross product of K hat in J hat is negative I hat. W we, that was a mouthful, wasn't it? So let's start to solve this problem finally. So first off, let us note that I had is in the positive X direction and that J hat is in the positive Y direction and that K hat is in the positive Z direction. Now, we must recall and use the cyclic property of unit vectors which states that in here to separate it, we'll write it in blue. So the cyclic property states that I had I should say the cross product of I hat, an eye hat is equal to zero. The cross product of I hat and J hat is equal to K hat. And the cross products of I hat in K hat is equal to negative J hat. And now following the same process J hat or I should say the cross product of J hat and I had is equal to negative K hat. Cross products of J hat and J hat is equal to zero. And finally cross products with Jay hat and a hat is equal to I hat. And now for K so K hat or the cross product of K hat and I hat is equal to J hat and the cross products of K hat and J hat is equal to negative I hat in the cross products of K hat in K hat is equal to zero. OK? So this is a lot to remember at first. So if you're freaking out, don't panic, there's a simple way to remember this. OK. I'll go back to you. Actually, we'll, we'll go back to using black. So a quick little diagram to help us remember this is if we just write down I hat, J hat NK hat in a circle shape. So I need to do just follow the arrow. So to remember the cross product of eye hat, NJ hat. So if you cross multiply I hat and J hat, you'll get K hat. Let's write that down. So the cross product of I hat and J hat is equal to K hat, which is the same. Is there a list of cross products that we just went over together and same process. So let's do the cross product of J hat in K hat. So the cross product of J hat in K hat will give us I hat and that is the same. So J hat multiply or I should say cross product of J hat and K hat will give us I hat. So that's the same as our cross product that we wrote down and et cetera. So then we can do it one last time, make sure we understand what's going on. So the cross product of khat, then I hat equals J hat. So K hat following or arrow, I'll do it in red. So K hat, the cross rod A K hat and I hat will give us J hat. So you just follow the arrows to get the motion, right? So that's a quick way to remember all of this. And then also it's super easy to remember that A, that A I hat multiplied by I hat. So like they're the unit vector multiplied by itself, the same value will give us zero. So that's an easy one to remember. And then all the other ones, all you have to do is just memorize this quick little diagram and then it's a breeze. It's like we'll go by in a snap. OK. So with that information in mind, we know that the correct answer focusing on the three cross products that we were supposed to find given to us in the prompt. The correct answer has to be the letter C which states that J, the cross product of J hat and I hat will give us negative K hat and the cross product of K hat and I hat will give us J hat and the cross product of K hat and J hat will give us negative I hat. OK. And that's it. Thank you so much for watching. Hopefully that helped and I can't wait to see you in the next video. Bye.

Struggling with Physics?

(I) Show that (b) î x ĵ, = k̂ , î x k̂ , = - ĵ , and ĵ x k̂ = î .

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