The Theoretical Minimum book question

[OrigamiCaptain]

I'm reading the book The Theoretical Minimum. I wonder if two one of the problems in the book can be thought of graphically. The mathematical solution didn't even occur to me, although it make perfect sense and probably should have been obvious.

1. Can you explain why the dot product of two vectors that are orthogonal is 0?

2. A dot B= abs(a)abs(b)cosθ

I know now that if it is orthogonal than θ=∏/2 and cos∏/2=0

Is there a way of visualizing i, j and k to get the same answer?3. For a second I thought that connecting x, y z together end to end would create a situation that would eliminate distance so the magnitude would equal zero, but after thinking about it a little more before posting this I don't believe that would work. I actually feel like there is something to this line of think though, which is why I'm asking this question.

Thank you for your time and consideration

 

[HallsofIvy]

The dot product of two vectors can be thought of as the projection of one vector on the other- that's where the |b|cos(θ) comes from. If two vectors are orthogonal then the projections are of 0 length.

 

[OrigamiCaptain]

The dot product of two vectors can be thought of as the projection of one vector on the other- that's where the |b|cos(θ) comes from. If two vectors are orthogonal then the projections are of 0 length.

I think I get it, but could you be a little more specific what you mean by projection?

Thank you for the help.

 

[olivermsun]

The projection is like the shadow of an object on the ground (with the sun directly overhead).
If you are orthogonal (perpendicular) to the plane of the ground (and very, very skinny) then your shadow should be 0.

 

[OrigamiCaptain]

Got it! Thanks!