Example 1-7-14 - Maple Help

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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

$V = A - B$ = $[\begin{array}{c} 5 \\ 1 \\ 7 \end{array}] - [\begin{array}{c} 0 \\ 0 \\ 4 / 3 \end{array}]$ = $[\begin{array}{c} 5 \\ 1 \\ 17 / 3 \end{array}]$

•	The component of V orthogonal to N is $V_{⊥ N}$ , which is then

$V - \frac{V \cdot N}{N \cdot N} N$ = $[\begin{array}{c} 5 \\ 1 \\ 17 / 3 \end{array}] - \frac{24}{14} [\begin{array}{c} 1 \\ 2 \\ 3 \end{array}]$ = $[\begin{array}{c} 23 / 7 \\ - 17 / 7 \\ 11 / 21 \end{array}]$

so the projection of A onto $P$ is $B + V_{⊥ N}$ = $[\begin{array}{c} 0 \\ 0 \\ 4 / 3 \end{array}] + [\begin{array}{c} 23 / 7 \\ - 17 / 7 \\ 11 / 21 \end{array}]$ = $\frac{1}{7} [\begin{array}{c} 21 \\ - 17 \\ 13 \end{array}]$ .

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

$x + 2 y + 3 z = 4$ $\overset{make plane}{\to}$ $<< Plane 1 >>$ $\overset{assign to a name}{\to}$ $P$

Define point A

•	Write a list of the coordinates for point A.

•	Context Panel: Assign to a Name≻A

$[5, 1, 7]$ $\overset{assign to a name}{\to}$ $A$

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

$A, P$ = $[5, 1, 7], << Plane 1 >>$ $\overset{projection}{\to}$ $[\frac{23}{7}, - \frac{17}{7}, \frac{13}{7}]$

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

$P$ $\overset{normal}{\to}$ $\overset{assign to a name}{\to}$ $N$

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≻ $L$

$A, N$ $\overset{make line}{\to}$ $<< Line 1 >>$ $\overset{assign to a name}{\to}$ $L$

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

$L, P$ $\overset{intersection}{\to}$ $[\frac{23}{7}, - \frac{17}{7}, \frac{13}{7}]$

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

$⟨0, 0, 4 / 3⟩$ $\overset{assign to a name}{\to}$ $B$

•	Write the name of point A.

•	Context Panel: Evaluate and Display Inline

•	Context Panel: Conversions≻Column Vector

•	Context Panel: Assign to a Name≻a

$A$ = $[5, 1, 7]$ $\overset{to Vector}{\to}$ $\overset{assign to a name}{\to}$ $a$

Obtain V, the vector from point B to point A

•	Context Panel: Assign Name

$V = a - B$ $\overset{assign}{\to}$

Obtain VN, the projection of V onto the normal N

•	Write the sequence of names V and N

•	Context Panel: Evaluate and Display Inline

•	Context Panel: Student Multivariate Calculus≻Lines & Planes≻Projection

•	Context Panel: Assign to a Name≻VN

$V, N$ = $\overset{projection}{\to}$ $\overset{assign to a name}{\to}$ $VN$

Add to the position vector B, the component of V orthogonal to N

•	Write the expression for $B + V_{⊥ N}$

•	Context Panel: Evaluate and Display Inline

$B + (V - VN)$ =

Of course, the components of V along and orthogonal to N can be computed at a grainier level:

$B + (V - \frac{V \cdot N}{N \cdot N} N)$ =

Maple Solution - Coded

Project point A onto plane $P$

•	Install the Student MultivariateCalculus package.

$with (Student :- MultivariateCalculus) &colon;$

•	Use the Plane command to define plane $P$ .

$P ≔ Plane (x + 2 y + 3 z = 4) &colon;$

•	Use the Projection command to project A onto $P$ .

$Projection ([5, 1, 7], P)$ = $[\frac{23}{7}, - \frac{17}{7}, \frac{13}{7}]$

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.

$N ≔ GetNormal (P) &colon;$

•	Use the Line command to obtain the line through A and along N.

$L ≔ Line ([5, 1, 7], N) &colon;$

•	Apply the GetIntersection command to intersect line $L$ with plane $P$ .

$GetIntersection (L, P)$ = $[\frac{23}{7}, - \frac{17}{7}, \frac{13}{7}]$

<< Previous Example Section 1.7 Next Chapter >>

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