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Boundless Calculus
Advanced Topics in Single-Variable Calculus and an Introduction to Multivariable Calculus
Vectors and the Geometry of Space
Calculus Textbooks Boundless Calculus Advanced Topics in Single-Variable Calculus and an Introduction to Multivariable Calculus Vectors and the Geometry of Space
Calculus Textbooks Boundless Calculus Advanced Topics in Single-Variable Calculus and an Introduction to Multivariable Calculus
Calculus Textbooks Boundless Calculus
Calculus Textbooks
Calculus
Concept Version 9
Created by Boundless

Vectors in the Plane

Vectors are needed in order to describe a plane and can give the direction of all dimensions in one vector equation.

Learning Objective

  • Calculate the directions of the normal vector and the directional vector of a reference point


Key Points

    • In order to adequately describe a plane, you need more than a point—you need a normal vector.
    • The normal vector is perpendicular to the directional vector of the reference point.
    • You can find the equation of a vector that describes a plane by using the following equation: a(x−x0)+b(y−y0)+c(z−z0)=0a (x-x_0) + b ( y-y_0) + c(z-z_0)=0a(x−x​0​​)+b(y−y​0​​)+c(z−z​0​​)=0 .

Terms

  • vector

    a directed quantity, one with both magnitude and direction; the signed difference between two points

  • normal

    a line or vector that is perpendicular to another line, surface, or plane


Full Text

Vectors in the Plane

Planes in a three dimensional space can be described mathematically using a point in the plane and a vector to indicate its "inclination".

Normal Vector to a Plane

This plane may be described parametrically as the set of all points of the formR=R0+sV+tW\mathbf R = \mathbf {R}_0 + s \mathbf{V} + t \mathbf{W}R=R​0​​+sV+tW, where sss and ttt range over all real numbers, V\mathbf{V}V and W\mathbf{W}W are given linearly independent vectors defining the plane, and R0\mathbf{R_0}R​0​​ is the vector representing the position of an arbitrary (but fixed) point on the plane. The vectors V\mathbf{V}V and W\mathbf{W}W can be visualized as vectors starting at R0\mathbf{R_0}R​0​​ and pointing in different directions along the plane. Note that V\mathbf{V}V and W\mathbf{W}W can be perpendicular but not parallel.

General form of the equation of the plane

In order to find the equation of the plane, consider the following: Let r0\mathbf{r}_0r​0​​ be the position vector of some point P0=(x0,y0,z0)\mathbf{P_0} = (x_0,y_0,z_0)P​0​​=(x​0​​,y​0​​,z​0​​) , and let n=(a,b,c)\mathbf{n} = (a,b,c)n=(a,b,c) be a nonzero vector.

The plane determined by this point and vector consists of those points PPP , with position vector r\mathbf{r}r, such that the vector drawn from P0P_0P​0​​ to PPP is perpendicular to n\mathbf{n} n. Recall that two vectors are perpendicular if and only if their dot product is zero. As such, the equation that describes the plane is given by:

n⋅(r−r0)=0\mathbf{n} \cdot (\mathbf{r}-\mathbf{r}_0)=0n⋅(r−r​0​​)=0

We can expand this equation in terms of its components to give:

a(x−x0)+b(y−y0)+c(z−z0)=0a (x-x_0)+ b(y-y_0)+ c(z-z_0)=0a(x−x​0​​)+b(y−y​0​​)+c(z−z​0​​)=0

which we call the point-normal equation of the plane and is the general equation we use to describe the plane.

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