Chapter 4: Partial Differentiation
Section 4.8: Unconstrained Optimization
If both U=3 i−5 j+7 k and V=6 i+j−13 k are bound to the origin, project U onto V. Hint: Find the minimum distance from the tip of U to the line along V.
The distance d from the tip of U to the line along V is U−R, the norm of the vector U−R, where R=t V is the position-vector form for the line along L.
d=U−t V =3−57−t 61−13 = 206⁢t2+156⁢t+83
Minimize d by solving d′=0 for t^=−39103, so that t^V=1103−234−39507.
Of course, the projection of U onto V is also given by U·VV·V V, but
U·VV·V = 1V2 3−57·61−13 = −78206= −39103=t^
Maple Solution - Interactive
Student Multivariate Calculus
Obtain the line through the origin and along V
Write a sequence of the origin and the vector V.
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Line
Context Panel: Assign to a Name≻L
0,0,0,6,1,−13→make line<< Line 2 >>→assign to a nameL
Obtain the projection of U onto the line along V
Write a sequence of the point at the tip of U, and the name L, then press the Enter key.
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Projection
Context Panel: Conversions≻Column Vector
3,−5,7,L = 3,−5,7,<< Line 2 >>→projection−234103,−39103,507103→to Vector−234103−39103507103
Now, obtain this same projection by minimizing the distance from the tip of U to the line along V.
Obtain the position-vector form for line L
Write the name L.
Context Panel: Evaluate and Display Inline
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Representation
Context Panel: Assign to a Name≻R
L = << Line 2 >>→representation6⁢tt−13⁢t→assign to a nameR
Obtain d, the distance from U to line L
Write the difference of vectors U and R.
Press the Enter key.
Context Panel: Norm≻Euclidean
Context Panel: Simplify≻Assuming Real
Context Panel: Assign to a Name≻d
→assign to a name
Calculus palette: Differentiation operator
Press the Enter key.
Context Panel: Solve≻Solve
Context Panel: Assign to a Name≻T
ⅆⅆ t d=0
Evaluate R at the minimizing value of t
Expression palette: Evaluation template
Rx=a|f(x)T = −234103−39103507103
Maple Solution - Coded
Install the Student MultivariateCalculus package.
Define the vectors U and V.
Apply the Projection command to obtain the projection of U onto V.
Apply the simplify command to U−t V, using the Norm command from Student LinearAlgebra.
d≔simplifyStudent:-LinearAlgebra:-NormU−t V = 206⁢t2+156⁢t+83
Find the critical number for d by applying the diff and solve commands.
Obtain the projection of U onto V by computing the vector τ V.
τ V = −234103−39103507103
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