# Simple Vector / Plane question

1. Oct 28, 2011

### ZedCar

1. The problem statement, all variables and given/known data
If (3,2,2) are the Cartesian components of vector a and (2,2,1) are the Cartesian coordinates of a point Q, calculate the distance of a plane through point Q and normal to vector a from the origin.

State whether the plane is in the direction of a from the origin or not.

2. Relevant equations

(° = dot product)

p = n̂ ° r

3. The attempt at a solution

a = (3,2,2)
therefore magnitude of a = √17
unit vector of a = 1/√17 (3,2,2)

p = n̂ ° r
p = 1/√17 (2,2,1) ° (3,2,2)
p = 1/√17 (6,4,2)

The second part of the question above asks "State whether the plane is in the direction of a from the origin or not." Wasn't sure how to do this. If the question was about them being orthogonal I'd have used dot product = 0.

Last edited: Oct 28, 2011
2. Oct 29, 2011

### I like Serena

Hi ZedCar!

You have not calculated the dot product properly yet.
You should also sum the products of the components.

The resulting number can be positive or negative.
What would be the meaning if it is negative?

3. Oct 29, 2011

### HallsofIvy

Staff Emeritus
The nearest point on the plane to the origin is, geometrically, where a line through the origin perpendicular to the plane crosses the plane. So one way to solve this problem is to find where that line crosses the plane.

And, as I like Serena suggests, the question "State whether the plane is in the direction of a from the origin or not" is not one of perpendicularity or not- that's always true. The question is whether the given a points from the origin to the plane or the opposite direction: from the origin away from the plane.

4. Oct 29, 2011

### ZedCar

Thanks guys!

So the fact that the dot product answer is a positive value indicates that the plane is in the direction of a from the origin.

Do you mean, the dot product answer should be;

(6/√17 , 4/√17, 2/√17)

5. Oct 29, 2011

### I like Serena

No.
The dot product is defined as:
$$(a,b,c) \cdot (x,y,z) = ax+by+cz$$

And since your problem asks for a distance, the result should not be a vector but a number.

6. Oct 29, 2011

### ZedCar

Okay.

I'm getting then, 12/√17 for the distance answer.

The fact that this answer is a positive value indicates that the plane is in the direction of a from the origin?

7. Oct 29, 2011

### I like Serena

Yep!

8. Oct 29, 2011

### ZedCar

Thanks very much!