Finding planes and angels between them?

[Prettygoodmilkshake]

Hi, I have exam next week and I am studying the book Mathematical Methods for Physicists by Tai Chow. I am trying to solve its problems for exercise but questions like these confuses me;

I thought I knew the basics but I freeze and cant make progress when I try to solve problems like these two. Unfortunately the book hasnt got solutions. I need to figure this out before the exam. Any suggestion will be appreciated.

 

[Dr.Peterson]

Hi, I have exam next week and I am studying the book Mathematical Methods for Physicists by Tai Chow. I am trying to solve its problems for exercise but questions like these confuses me;
View attachment 31889View attachment 31888
I thought I knew the basics but I freeze and cant make progress when I try to solve problems like these two. Unfortunately the book hasnt got solutions. I need to figure this out before the exam. Any suggestion will be appreciated.

What DO you know that you can use? Can you find the normal vector to a surface? Can you find the normal vector to a plane from two vectors in it?

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[Prettygoodmilkshake]

For the first question. I found this first:
then similarly found the other one;
and used them to find cosdelta.
But I thought its not true since I cannot find the angle from this cos value.

For the second question I couldnt do anything beacuese I couldnt figure out how to find the surface with these two (A, B) given points. If I can define the surface then I know how to find a normal vector to it. Thanks for your help.

 

[Dr.Peterson]

But I thought its not true since I cannot find the angle from this cos value.

Why do you say that? I wonder if you miscalculated the value, and got something greater than 1. It is not.

For the second question I couldnt do anything beacuese I couldnt figure out how to find the surface with these two (A, B) given points. If I can define the surface then I know how to find a normal vector to it. Thanks for your help.

If A and B are points, then you are right; there is not one place defined by two points. I have to interpret the problem as saying those are two vectors. Is it possible that that is what was intended? Can you show us the actual problem as an image?

If I'm right, then a cross product can be useful.

 

[Prettygoodmilkshake]

Unfortunately this is the actual problem I didnt cut it. Oh, I didnt think those as vectors since notation looked like points to me. Probably you are right. Then all I need to do just cross product them and divide it by its lenght. Thank you again!