# Find the vector and cartesian equations of a plane

• ~angel~
In summary: You might also check that the normal vector is perpendicular to the other vector given, just in case you made a mistake somewhere.)In summary, the conversation involves solving equations and finding the equations of lines and planes. Specific topics discussed include finding parallel and perpendicular lines and planes, finding normal vectors, and finding the angle between two lines.
~angel~
These are just a few questions that a don't understand and any help would be great.

1. Prove that the line

(x-3)/2 = (y-4)/3 = (z-5)/4

is parallel to the plat 4x + 4y - 5z = 14

2. Find the equation of the line through (1,0,-2) and perpendicular to the plane
3x - 4y + z -6 = 0

I'm assuming you need to find the normal of the plane, but I'm not sure how to do that.

3. This is the 2nd part of a question: Find the cosine of the angle between the directions of the line in (a) and the line

(x+2)/2 = y/3 = (z-1)/3

The line in (a) is (x-1)i + (y-1)j + (z-2)k = t (3i + j + k), which becomes the cartesian equation:

(x-1)/3 = (y-1)/1 = (z-2)/1

4. Find the vector and cartesian equations of a plane containing the line

(x-4)/-2 = (y+3)/1 = (z-1)/3

I know all the basic things in vectors, but these are a few questions I just want to clear up due to my upcoming exam.

Any help would be greatly appreciated.

For a plane

$$ax+by+cz = d$$

The normal vector can be denoted by

$$\vec{N} = <a,b,c>$$

~Angel~ said:
1. Prove that the line

(x-3)/2 = (y-4)/3 = (z-5)/4

is parallel to the plat 4x + 4y - 5z = 14
As whozum told you, a vector perpendicular to the plane is <4, 4,-5>.
A vector in the direction of the <2, 3, 4> (Set each of those fractions equal to the parameter t and solve for x, y, z). The line will be parallel to the plane if these two vectors are pependicular to one another.

2. Find the equation of the line through (1,0,-2) and perpendicular to the plane
3x - 4y + z -6 = 0

I'm assuming you need to find the normal of the plane, but I'm not sure how to do that.
Whozum's suggestion again. The components of the vector perpendicular to the plane are just the coefficients of the variables.

3. This is the 2nd part of a question: Find the cosine of the angle between the directions of the line in (a) and the line

(x+2)/2 = y/3 = (z-1)/3

The line in (a) is (x-1)i + (y-1)j + (z-2)k = t (3i + j + k), which becomes the cartesian equation:

(x-1)/3 = (y-1)/1 = (z-2)/1
So you know that a vector in the direction of the line in (a) is <3, 1, 1> and a vector in the direction of the other line is <3, 1, 1>. What do you think the angle between them is?

4. Find the vector and cartesian equations of a plane containing the line

(x-4)/-2 = (y+3)/1 = (z-1)/3

Notice "of a plane". There are an infinite number of planes containing any line- you are only asked to find one. Again, the vector <-2, 1, 3> is in the direction of the line. Any plane containing the line has normal vector perpendicular to that.
You just need to find one vector perpendicular to that and then be sure the plane contains that line.

## 1. What is a vector and cartesian equation of a plane?

A vector equation of a plane is a representation of the plane using a vector and a point. It can be written as r = a + sb + tc, where r is a position vector, a is a point on the plane, and b and c are direction vectors. A cartesian equation of a plane is a representation of the plane using its normal vector and a point. It can be written as Ax + By + Cz = d, where A, B, and C are the components of the normal vector and d is a constant.

## 2. How do I find the normal vector of a plane?

The normal vector of a plane can be found by taking the cross product of two non-parallel vectors on the plane. These vectors can be found by subtracting one point from another on the plane. The resulting vector will be orthogonal to the plane and can be used as the normal vector in the cartesian equation.

## 3. Can a plane have multiple vector and cartesian equations?

Yes, a plane can have an infinite number of vector and cartesian equations. This is because there are infinite combinations of points and direction vectors that can be used to represent a plane.

## 4. How do I convert between vector and cartesian equations of a plane?

To convert from a vector equation to a cartesian equation, you can use the formula Ax + By + Cz = d, where A, B, and C are the components of the direction vectors and d is a constant. To convert from a cartesian equation to a vector equation, you can use the formula r = a + sb + tc, where a is a point on the plane and b and c are the direction vectors.

## 5. How can I use the vector and cartesian equations of a plane in real-life applications?

The vector and cartesian equations of a plane are used in various fields such as engineering, physics, and computer graphics. They are used to represent and calculate the position and direction of objects in 3D space. For example, in engineering, the equations can be used to design and analyze structures such as bridges and buildings. In computer graphics, they are used to create 3D models and animations.

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