# Finding the vector equation of a plane

• i_love_science
In summary, the conversation discusses a vector equation with the given vectors u = [-2,3,1], v = [-2,2,3], and the points Po = (6,0,0) and P = (4,2,3). The solution involves finding the values of r and q, which are real numbers, in the equation [6,0,0] + r[-2,3,1] + q[-2,2,3]. There is a slight difference in the answer given by the person asking the question, but it is shown that their answer is transformable to the correct solution.
i_love_science
Homework Statement
Find a vector equation of the plane that passes through the point (6, 0, 0) and contains the line 𝑥 = 4 - 2𝑡, 𝑦 = 2 + 3𝑡, 𝑧 = 3 + 𝑡.
Relevant Equations
vector equation
Solution:
u = [-2,3,1]
Po = (6,0,0) & P = (4,2,3)
PoP = v = [-2,2,3]
Therefore, the answer is [6,0,0] + r[-2,3,1] + q[-2,2,3]; r, q are real numbers

I don't understand why (6,0,0) is used as the point in the vector equation, since it only lies on the [-2,2,3] vector, not the u = [-2,3,1] vector.

My answer is r = [4,2,3] + r[2,-2,-3] + q[-2,3,1], r, q are real numbers

Could anyone explain the solution, and is my answer correct? Thanks.

$$[4,2,3]+q[-2,3,1]-r[-2,2,3]=[6,0,0]+q[-2,3,1]-(r-1)[-2,2,3]$$
changing q with r and -(r-1) with q, it equals to the text answer.

i_love_science

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

A vector equation of a plane is a mathematical representation of a plane using a vector and a point on the plane. It is in the form of r = r0 + sa + tb, where r is a position vector, r0 is a point on the plane, and a and b are two non-parallel vectors that lie on the plane.

## 2. How do you find the vector equation of a plane?

To find the vector equation of a plane, you need to know a point on the plane and two non-parallel vectors that lie on the plane. You can use these three pieces of information to form the equation r = r0 + sa + tb, where r is the position vector, r0 is the known point, and a and b are the two vectors.

## 3. What is the purpose of the vector equation of a plane?

The vector equation of a plane is used to describe the position of points on a plane in a three-dimensional space. It is also useful in solving problems involving planes, such as finding intersections between planes or finding the angle between two planes.

## 4. Can the vector equation of a plane be written in different forms?

Yes, the vector equation of a plane can be written in different forms, such as r = r0 + sa + tb or r = r0 + sa + ta + ub. The important thing is that the equation represents a point on the plane and two non-parallel vectors that lie on the plane.

## 5. How is the vector equation of a plane used in real-life applications?

The vector equation of a plane is used in various real-life applications, such as computer graphics, engineering, and physics. It is used to describe the position and orientation of objects in a three-dimensional space, and it is also used in calculating forces and motion in a plane. In addition, it is used in navigation systems to determine the position of an object relative to a plane.

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