Question

5. Find the minimum distance between the point Q: (4,0,1) and a point on the line \( \vec{r} = \begin{bmatrix} 1 \ -4 \ 2 \end{bmatrix} + t \begin{bmatrix} 1 \ 2 \ -1 \end{bmatrix} \). Use the (not-to-scale) diagram below to get you started. (a) Do it using a combination of vectors and calculus by writing an expression for the distance between Q and any point on the line, then minimizing t. (b) Do it using vectors only by finding the magnitude of a vector perpendicular to the line that ends on Q. Q: (4,0,1) P: (1,-4,2) \( \vec{m} = \begin{bmatrix} 1 \ 2 \ -1 \end{bmatrix} \)

          5. Find the minimum distance between the point Q: (4,0,1) and a point on the line \( \vec{r} = \begin{bmatrix} 1 \ -4 \ 2 \end{bmatrix} + t \begin{bmatrix} 1 \ 2 \ -1 \end{bmatrix} \). Use the (not-to-scale) diagram below to get you started.
(a) Do it using a combination of vectors and calculus by writing an expression for the distance between Q and any point on the line, then minimizing t.
(b) Do it using vectors only by finding the magnitude of a vector perpendicular to the line that ends on Q.
Q: (4,0,1)
P: (1,-4,2)
\( \vec{m} = \begin{bmatrix} 1 \ 2 \ -1 \end{bmatrix} \)

5. Find the minimum distance between the point Q: (4,0,1) and a point on the line r⃗ =
< b m a t r i x >
+ t
< b m a t r i x >. Use the (not-to-scale) diagram below to get you started.
(a) Do it using a combination of vectors and calculus by writing an expression for the distance between Q and any point on the line, then minimizing t.
(b) Do it using vectors only by finding the magnitude of a vector perpendicular to the line that ends on Q.
Q: (4,0,1)
P: (1,-4,2)
m⃗ =
< b m a t r i x >

Added by Joshua F.

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