Simple Vector / Plane question

[ZedCar]

Homework Statement

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.

Homework Equations

(° = dot product)

p = n̂ ° r

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.

 

[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?

 

[HallsofIvy]

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.

 

[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.

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

Do you mean, the dot product answer should be;

(6/√17 , 4/√17, 2/√17)

 

[I like Serena]

Do you mean, the dot product answer should be;

(6/√17 , 4/√17, 2/√17)

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.

 

[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?

 

[I like Serena]

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?

Yep!

 

[ZedCar]

Thanks very much!