Chapter 1: Vectors, Lines and Planes
Section 1.7: Planes
Example 1.7.14
Project point A:5,1,7 onto P, the plane described by the equation x+2 y+3 z=4.
Solution
Mathematical Solution
Figure 1.7.14(a) shows point A as a black dot, and point B as a gold dot.
Vector V from B to A is shown as a red arrow.
The normal N is shown as the blue arrow, and the component of V orthogonal to N is shown as the green arrow.
Plane P, also shown in the figure, is transparent, so position vector B, represented by the short black arrow, actually lies below the plane.
The projection of point A onto plane P is the point at which the vector N has been drawn. Its coordinates are the head of the vector B+V⊥N.
Figure 1.7.14(a) Vectors V (red), N (blue), and V⊥N (green)
Point B is an arbitrary point on plane P, and is taken as 0,0,4/3 for convenience. Hence
V=A−B = 517−004/3 = 5117/3
The component of V orthogonal to N is V⊥N , which is then
V−V·NN·N N = 5117/3−2414123 = 23/7−17/711/21
so the projection of A onto P is B+V⊥N = 004/3+23/7−17/711/21 = 1721−1713.
Maple Solution - Interactive
Tools≻Load Package: Student Multivariate Calculus
Loading Student:-MultivariateCalculus
Define plane P
Control-drag the equation of the plane.
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Plane
x+2 y+3 z=4→make plane<< Plane 1 >>→assign to a nameP
Define point A
Write a list of the coordinates for point A.
Context Panel: Assign to a Name≻A
5,1,7→assign to a nameA
Obtain the projection of point A on plane P
Write a sequence of the names for the point and the plane.
Context Panel: Evaluate and Display Inline
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Projection
A,P = 5,1,7,<< Plane 1 >>→projection237,−177,137
There are two traditional approaches possible.
The following approach obtains the equation of the line that is through point A and that has direction N, the normal to the plane, and intersects this line with the plane.
Obtain N, the normal for plane P
Write the name of the plane.
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Normal
Context Panel: Assign to a Name≻N
P→normal →assign to a nameN
Define L, the line through point A in the direction of N
Write the sequence of names for the point A and the normal N
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Line
Context Panel: Assign to a Name≻L
A,N→make line<< Line 1 >>→assign to a nameL
Obtain the intersection of line L with plane P
Write the sequence of names for the line L and the plane P
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Intersection
L,P→intersection237,−177,137
The following approach is vector-based. An arbitrary point, say B:0,0,4/3, is selected in the plane and the component of V, the vector from B to A, orthogonal to N is added to the position vector B.
Define the position vectors B and a
Context Panel: Assign to a Name≻B
0,0,4/3→assign to a nameB
Write the name of point A.
Context Panel: Conversions≻Column Vector
Context Panel: Assign to a Name≻a
A = 5,1,7→to Vector →assign to a namea
Obtain V, the vector from point B to point A
Context Panel: Assign Name
V=a−B→assign
Obtain VN, the projection of V onto the normal N
Write the sequence of names V and N
Context Panel: Assign to a Name≻VN
V,N = →projection →assign to a nameVN
Add to the position vector B, the component of V orthogonal to N
Write the expression for B+V⊥N
B+V−VN =
Of course, the components of V along and orthogonal to N can be computed at a grainier level:
B+V−V·NN·NN =
Maple Solution - Coded
Project point A onto plane P
Install the Student MultivariateCalculus package.
withStudent:-MultivariateCalculus:
Use the Plane command to define plane P.
P≔Planex+2 y+3 z=4:
Use the Projection command to project A onto P.
Projection5,1,7,P = 237,−177,137
Obtain line L through A and along N, and intersect it with plane P
Apply the GetNormal command to plane P to obtain normal N.
N≔GetNormalP:
Use the Line command to obtain the line through A and along N.
L≔Line5,1,7,N:
Apply the GetIntersection command to intersect line L with plane P.
GetIntersectionL,P = 237,−177,137
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