Finding the equation of a plane ("A golf ball is hit from a tee towards a hole...")

[tpupble]

I am having difficulty comprehending part b of the below question
From what I understand of finding a plane’s equation, you need three points which you will sub into the formula r⋅n = a⋅n.

r is (x,y,z)
n is the normal vector formed by the cross product of two lines originating from one common point.
a is the common point.

In this scenario there is only two - the tee/origin (0,0,0) and the hole (250, 25°, 2°) ≈ (226.44, 105.59, 8.73). How should I proceed?

 

[topsquark]

I am having difficulty comprehending part b of the below questionView attachment 36445
From what I understand of finding a plane’s equation, you need three points which you will sub into the formula r⋅n = a⋅n.

r is (x,y,z)
n is the normal vector formed by the cross product of two lines originating from one common point.
a is the common point.

In this scenario there is only two - the tee/origin (0,0,0) and the hole (250, 25°, 2°) ≈ (226.44, 105.59, 8.73). How should I proceed?

You have a function r(t). Pick a t, any t...

Otherwise, you know the base line of the plane (along the ground). The plane will go straight up.

-Dan

 

[blamocur]

I am having difficulty comprehending part b of the below questionView attachment 36445
From what I understand of finding a plane’s equation, you need three points which you will sub into the formula r⋅n = a⋅n.

r is (x,y,z)
n is the normal vector formed by the cross product of two lines originating from one common point.
a is the common point.

In this scenario there is only two - the tee/origin (0,0,0) and the hole (250, 25°, 2°) ≈ (226.44, 105.59, 8.73). How should I proceed?

I agree that there is not enough data there. I wonder if the authors of the problem meant the slope is the steepest in the direction between the tee and the hole. I.e., if the plane is horizontal in the orthogonal direction.

 

[topsquark]

Whoops! Please disregard my post. I misread the question.

-Dan

 

[tpupble]

Thanks for having a look. I peeked at the worked solution attached below and they have formed the normal vector by adding 90° to the depth (Z axis). Should I do this whenever there is only two points and why is it that the magnitude of the vector can be arbitrary?

 

[blamocur]

Should I do this whenever there is only two points...

IMHO, the statement is incomplete, i.e. the book authors did a sloppy job.

... why is it that the magnitude of the vector can be arbitrary?

two vectors with the same direction but different magnitudes sill define the same (infinite) line, which is used in the definition the plane. Another way to look at it: if vector $\mathbf v$ lies in a plane, then all vectors $\alpha \mathbf v$ (where $\alpha$ is an arbitrary scalar) will lie in the same plane.