Finding a vector perpendicular to the plane 2x + y - z = 7 (answer is -4i - 2j + 2k)

[tpupble]

A plane is represented in vector form as 2x + y - z = 7. A vector perpendicular to the plane is…

the answer to this multiple choice question is -4i - 2j + 2k but I do not understand how this is so. I thought that for a vector to be perpendicular, its dot product with the vector orthogonal to the plane is 0. The vector orthogonal to the plane is 2i + j - k? A dot product with the vector -4i - 2j + 2k yields -12.

 

[Steven G]

Can you give me a vector on the plane 2x + y - z = 7? Is <2, 1, -1> a vector on this plane??
It would be really nice of you if you could post the whole problem.

 

[Dr.Peterson]

A plane is represented in vector form as 2x + y - z = 7. A vector perpendicular to the plane is…

the answer to this multiple choice question is -4i - 2j + 2k but I do not understand how this is so. I thought that for a vector to be perpendicular, its dot product with the vector orthogonal to the plane is 0. The vector orthogonal to the plane is 2i + j - k? A dot product with the vector -4i - 2j + 2k yields -12.

Yes, 2i + j - k is a vector perpendicular (orthogonal) to the given plane. So is any scalar multiple of that vector!

Which of the given choices is such a multiple?

(And when a question is multiple-choice, the list of choices is an essential part of the problem, so you really need to show them all -- though in this case I don't need to see in order to see what the problem is asking for.)

 

[tpupble]

The complete question is:
A plane is represented in vector form as 2x + y - z = 7. A vector perpendicular to the plane is
A) i + j - k
B) 3i + 2j - 2k
C) 9i + 3j - 3k
D) -4i - 2j + 2k

thank you, I now get why it is D.

 

[Dr.Peterson]

The complete question is:
A plane is represented in vector form as 2x + y - z = 7. A vector perpendicular to the plane is
A) i + j - k
B) 3i + 2j - 2k
C) 9i + 3j - 3k
D) -4i - 2j + 2k

thank you, I now get why it is D.

Good.

Note that the wording, "A vector perpendicular to the plane is ..." tells you that they are not looking for the [only] such vector, but the one in the list that is a [multiple of the] normal vector, which is why the list of choices is, in general, an essential part of the problem -- and in some problems there could be more than one correct answer.