Vectors Linear Algebra questions

[xax129]

Howdie,
so I am in college in Linear Algebra and we are currently seeing vectors. I am someone who understands math with examples, and he did not give us any for the 2 following questions.
Start with the vectorial equation of a plane in

;

with

.

Give a normal (cartesian) equation of the plane

where

is a normal vector obtained by computing

(but any vectors parallel to it would normally do the trick).

For this first questions, we would need to find A,B,C,D.



Second question

Start with the normal (cartesian) equation of a plane in

;

Let

be a point in

.

Compute

the distance between the plane and the point. Find

the closest point on the plane

to

.

Any help would be really appreciated as this is for Wednesday.

 

[HallsofIvy]

Howdie,
so I am in college in Linear Algebra and we are currently seeing vectors. I am someone who understands math with examples, and he did not give us any for the 2 following questions.
Start with the vectorial equation of a plane in

;

with

.

Give a normal (cartesian) equation of the plane

where

is a normal vector obtained by computing

(but any vectors parallel to it would normally do the trick).

For this first questions, we would need to find A,B,C,D.

DID you calculate that cross product to find a, b, and c? (not "A, B, C")
You can find d by setting x, y, z equal to some point on the plane which you can get by setting s and t equal to some numbers.

Second question

Start with the normal (cartesian) equation of a plane in

;

Let

be a point in

.

Compute

the distance between the plane and the point. Find

the closest point on the plane

to

.

Any help would be really appreciated as this is for Wednesday.

The C, on the plane, that is closest to the given point is on a line perpendicular to the plane. You know that, since the equation of the plane is 5x+ 2y- 3z= -20, that 5i+ 2j- 3k is a vector perpendicular to that plane. Find the equation of the line parallel to 5i+ 2j- 3z that passes through (-4, -4, 1). C is the point where that line crosses the plane.