# Calculus 3 : 3-Dimensional Space

## Example Questions

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### Example Question #1 : Equations Of Lines And Planes

Write down the equation of the line in vector form that passes through the points , and .

Explanation:

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 .

Explanation:

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 #1 : 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.

Explanation:

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 #4 : Equations Of Lines And Planes

Find the angle (in degrees) between the planes ,

Explanation:

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 #5 : Equations Of Lines And Planes

Determine the equation of the plane that contains the following points.

Explanation:

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 #6 : Equations Of Lines And Planes

Find the equation of the plane containing the points

Explanation:

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 #7 : Equations Of Lines And Planes

Which of the following is an equation of a plane parallel to the plane ?

Explanation:

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 #8 : Equations Of Lines And Planes

Find a parametric representation of the curve of intersection of the cylinder  and the plane .

Explanation:

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 #9 : Equations Of Lines And Planes

Find the equation of the plane containing the following points

Explanation:

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

Explanation:

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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