All Calculus 3 Resources
Example Questions
Example Question #1 : 3 Dimensional Space
Write down the equation of the line in vector form that passes through the points , and .
Remember the general equation of a line in vector form:
, where is the starting point, and is the difference between the start and ending points.
Lets apply this to our problem.
Distribute the
Now we simply do vector addition to get
Example Question #2 : Equations Of Lines And Planes
Find the approximate angle between the planes , and .
None of the other answers.
Finding the angle between two planes requires us to find the angle between their normal vectors.
To obtain normal vectors, we simply take the coefficients in front of .
The (acute) angle between any two vector is
,
Substituting, we have
.
Example Question #2 : Equations Of Lines And Planes
Find the point of intersection of the plane and the line described by
The line and the plane are parallel.
Substituting the components of the line into those of the plane, we have
Substituting this value of back into the components of the line gives us
.
Example Question #1 : Equations Of Lines And Planes
Find the angle (in degrees) between the planes ,
A quick way to notice the answer is is to notice the planes are parallel (They only differ by the constant on the right side).
Typically though, to find the angle between two planes, we find the angle between their normal vectors.
A vector normal to the first plane is
A vector normal to the second plane is
Then using the formula for the angle between vectors, , we have
.
Example Question #3 : Equations Of Lines And Planes
Determine the equation of the plane that contains the following points.
The equation of a plane is defined as
where is the normal vector of the plane.
To find the normal vector, we first get two vectors on the plane
and
and find their cross product.
The cross product is defined as the determinant of the matrix
Which is
Which tells us the normal vector is
Using the point and the normal vector to find the equation of the plane yields
Simplified gives the equation of the plane
Example Question #3 : Equations Of Lines And Planes
Find the equation of the plane containing the points
The equation of a plane is defined as
where is the normal vector of the plane.
To find the normal vector, we first get two vectors on the plane
and
and find their cross product.
The cross product is defined as the determinant of the matrix
Which is
Which tells us the normal vector is
Using the point and the normal vector to find the equation of the plane yields
Simplified gives the equation of the plane
Example Question #3 : Equations Of Lines And Planes
Which of the following is an equation of a plane parallel to the plane ?
None of the other answers
Planes that are parallel to each other only differ (if at all) by the constant on the right-hand side (when both sides are simplified). Since has the same coefficients as the given plane, they are parallel to each other.
Example Question #2 : 3 Dimensional Space
Find a parametric representation of the curve of intersection of the cylinder and the plane .
We can begin by rewriting the expression for the cylinder as follows
.
This tells us that . Plugging this back into the equation for the plane to find .
This gives us the representation of the curve of intersection as
.
Example Question #1 : Equations Of Lines And Planes
Find the equation of the plane containing the following points
The equation of a plane is defined as
where
is the normal vector of the plane.
To find the normal vector, we first get two vectors on the plane
and
and find their cross product.
The cross product is defined as the determinant of the matrix
Which is
Which tells us the normal vector is
Using the point
and the normal vector to find the equation of the plane yields
Simplified gives the equation of the plane
Example Question #10 : Equations Of Lines And Planes
Find the angle in degrees between the planes and
None of the other answers
To find the angle between the planes, we find the angle between their normal vectors.
We have
, for the first plane, and
, for the second plane.
The angle between these two vectors is
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