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Example Questions
Example Question #1 : Normal Vectors
Find the Unit Normal Vector to the given plane.
.
Recall the definition of the Unit Normal Vector.
Let
Example Question #1 : Normal Vectors
Find the unit normal vector of .
Does not exist
Does not exist
To find the unit normal vector, you must first find the unit tangent vector. The equation for the unit tangent vector, , is
where is the vector and
is the magnitude of the vector.
The equation for the unit normal vector,, is
where is the derivative of the unit tangent vector and
is the magnitude of the derivative of the unit vector.
For this problem
There is no unit normal vector of .
Example Question #3 : Normal Vectors
Find the unit normal vector of .
Does not exist
To find the unit normal vector, you must first find the unit tangent vector. The equation for the unit tangent vector, , is
where is the vector and
is the magnitude of the vector.
The equation for the unit normal vector,, is
where is the derivative of the unit tangent vector and
is the magnitude of the derivative of the unit vector.
For this problem
Example Question #4 : Normal Vectors
Find the unit normal vector of .
DNE
DNE
To find the unit normal vector, you must first find the unit tangent vector. The equation for the unit tangent vector, , is
where is the vector and
is the magnitude of the vector.
The equation for the unit normal vector,, is
where is the derivative of the unit tangent vector and
is the magnitude of the derivative of the unit vector.
For this problem
The normal vector of does not exist.
Example Question #5 : Normal Vectors
Find the unit normal vector of .
To find the unit normal vector, you must first find the unit tangent vector. The equation for the unit tangent vector, , is
where is the vector and
is the magnitude of the vector.
The equation for the unit normal vector,, is
where is the derivative of the unit tangent vector and
is the magnitude of the derivative of the unit vector.
For this problem
Example Question #6 : Normal Vectors
Find a normal vector that is perpendicular to the plane given below.
No such vector exists.
Derived from properties of plane equations, one can simply pick off the coefficients of the cartesian coordinate variable to give a normal vector that is perpendicular to that plane. For a given plane, we can write
.
From this result, we find that for our case,
.
Example Question #1 : Normal Vectors
Which of the following is FALSE concerning a vector normal to a plane (in -dimensional space)?
The cross product of any two normal vectors to the plane is .
Multiplying it by a scalar gives another normal vector to the plane.
It is parallel to any other normal vector to the plane.
All the other answers are true.
It is orthogonal to the plane.
All the other answers are true.
These are all true facts about normal vectors to a plane. (If the surface is not a plane, then a few of these no longer hold.)
Example Question #8 : Normal Vectors
Determine whether the two vectors, and
, are orthogonal or not.
The two vectors are not orthogonal.
The two vectors are orthogonal.
The two vectors are orthogonal.
Vectors can be said to be orthogonal, that is to say perpendicular or normal, if their dot product amounts to zero:
To find the dot product of two vectors given the notation
Simply multiply terms across rows:
For our vectors, and
The two vectors are orthogonal.
Example Question #9 : Normal Vectors
Determine whether the two vectors, and
, are orthogonal or not.
The two vectors are not orthogonal.
The two vectors are orthogonal.
The two vectors are not orthogonal.
Vectors can be said to be orthogonal, that is to say perpendicular or normal, if their dot product amounts to zero:
To find the dot product of two vectors given the notation
Simply multiply terms across rows:
For our vectors, and
The two vectors are not orthogonal.
Example Question #10 : Normal Vectors
Determine whether the two vectors, and
, are orthogonal or not.
The two vectors are not orthogonal.
The two vectors are orthogonal.
The two vectors are not orthogonal.
Vectors can be said to be orthogonal, that is to say perpendicular or normal, if their dot product amounts to zero:
To find the dot product of two vectors given the notation
Simply multiply terms across rows:
For our vectors, and
The two vectors are not orthogonal.
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