Proving the vector OA is normal to the plane OBC

In summary: Therefore, the cross product of OA and \vec n# is also a scalar multiple of ##\vec n##. Therefore, the cross product of OA and \vec n# is also zero. Therefore, OA is normal to the plane OBC.
  • #1
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Homework Statement



Given position vectors: OA ( i + 2j - k) ; OB ( -i + 2j +3k); OC ( 2i + j + 4k)

Given that OA is perpendicular to OB.

The Question : Show that OA is normal to the plane OBC.

Homework Equations


r . n = d

To find the normal of the plane OBC, I used n = BO x BC

d = OB . n

The Attempt at a Solution



Equation of plane OBC:

BO = -OB = i - 2j - 3k

BC = OC - OB = (2,1,4) - (-1,2,3) = (3,-1,1)

n = BO x BC = (5,10,-5)

d = OB . n = (-1,2,3) . (5,10,-5) = 0

∴ Equation of plane OBC = r. (5,10,-5) = 0


How do I show that OA is normal to the plane? I know that it is related to the statement - 'OA is perpendicular to OB'. However, I do not know how can it be applied into the question.
I hope someone can help me out.

Thank you for your time! :smile:
 
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  • #2
I agree with the normal vector to the plane OBC, that is [tex] \vec{n} = 5\vec{i} +10\vec{j} -5\vec{k}.[/tex]
You are given that OA is perpendicular to OB, which you can easily verify.
To show that OA is normal to the plane, show that the cross product of OA with [itex] \vec{n} [/itex] is 0.
 
  • #3
CAF123 said:
I agree with the normal vector to the plane OBC, that is [tex] \vec{n} = 5\vec{i} +10\vec{j} -5\vec{k}.[/tex]
You are given that OA is perpendicular to OB, which you can easily verify.
To show that OA is normal to the plane, show that the cross product of OA with [itex] \vec{n} [/itex] is 0.
Thank you very much for your help! :smile:
 
  • #4
CAF123 said:
I agree with the normal vector to the plane OBC, that is [tex] \vec{n} = 5\vec{i} +10\vec{j} -5\vec{k}.[/tex]
You are given that OA is perpendicular to OB, which you can easily verify.
To show that OA is normal to the plane, show that the cross product of OA with [itex] \vec{n} [/itex] is 0.

Or even easier, observe that OA is a scalar multiple of ##\vec n##.
 

1. How do you define a normal vector?

A normal vector is a vector that is perpendicular, or at a 90 degree angle, to a given surface or plane. It is commonly denoted by a lowercase "n" with an arrow above it.

2. What is the significance of proving a vector is normal to a plane?

Proving that a vector is normal to a plane is important because it provides crucial information about the relationship between the vector and the plane. It can also be used to solve various mathematical and physical problems involving vectors and planes.

3. How can you prove that a vector is normal to a plane?

There are several methods for proving that a vector is normal to a plane. One method is to show that the dot product of the vector with any other vector in the plane is equal to zero. Another method is to demonstrate that the cross product of the vector with any two vectors in the plane is equal to zero.

4. Why is the vector OA chosen for this proof?

The vector OA is chosen for this proof because it is the vector that connects the origin (O) to a point (A) on the plane. This vector is often used as the reference vector when proving that another vector is normal to the same plane.

5. Can a vector be normal to more than one plane?

Yes, a vector can be normal to more than one plane. This occurs when the vector is parallel to the line of intersection between two planes. In this case, the vector is normal to both planes.

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