Distance between a line parallel to a plane

Suppose you are given a line $\mathcal{L}: \ \mathbf{x}(t) = A + t \mathbf{v}$ parallel to a plane $\mathcal{P}: \mathbf{n} \cdot (X - B)$ . What is a procedure you could use to find the distance between the line $\mathcal{L}$ and the plane $\mathcal{P}$ ? Answer by dragging statements from the left column to the right column and putting them in the correct order in the right column. Note: your answer does not need to use all of the statements from the left column. Hint: draw a picture.

Statements to choose from: Drag these statements to the right column.

Find the vector $\mathbf{p} = \mathrm{proj}_{\mathbf{n}} \mathbf{AB}$ .
The desired distance is the length of $\mathbf{q}$ .
Construct the vector $\mathbf{AB}$ .
The desired distance is the length of $\mathbf{v}$ .
Find the vector $\mathbf{p} = \mathrm{proj}_{\mathbf{v}} \mathbf{AB}$ .
Find the vector $\mathbf{q} = \mathbf{AB} - \mathbf{p}$ .
The desired distance is the length of $\mathbf{p}$ .

Your solution: Put the statements in order in this column and press the Submit Answers button.

Statements to choose from: Drag these statements to the right column.

Find the vector $\mathbf{p} = \mathrm{proj}_{\mathbf{n}} \mathbf{AB}$ .
The desired distance is the length of $\mathbf{q}$ .
Construct the vector $\mathbf{AB}$ .
The desired distance is the length of $\mathbf{v}$ .
Find the vector $\mathbf{p} = \mathrm{proj}_{\mathbf{v}} \mathbf{AB}$ .
Find the vector $\mathbf{q} = \mathbf{AB} - \mathbf{p}$ .
The desired distance is the length of $\mathbf{p}$ .

Your solution: Put the statements in order in this column and press the Submit Answers button.

Distance between skew lines

Suppose you are given a line $\mathcal{L}_1 : \mathbf{x_1}(s) = A + s \mathbf{v}$ skew to another line $\mathcal{L}_2 : \mathbf{x_2}(t) = B + t \mathbf{w}$ . What is a procedure you could use to find the distance between these lines? Hint: draw a picture.

Statements to choose from: Drag these statements to the right column.

Find the vector $\mathbf{q} = \mathbf{AB} - \mathbf{p}$ .
The desired distance is the length of $\mathbf{p}$ .
The desired distance is the length of $\mathbf{w}$ .
Find the vector $\mathbf{p} = \mathrm{proj}_{\mathbf{n}} \mathbf{AB}$ .
The desired distance is the length of $\mathbf{q}$ .
Find the vector $\mathbf{p} = \mathrm{proj}_{\mathbf{v}} \mathbf{AB}$ .
Find the vector $\mathbf{n} = \mathbf{v} \times \mathbf{w}$ .

Your solution: Put the statements in order in this column and press the Submit Answers button.

Statements to choose from: Drag these statements to the right column.

Find the vector $\mathbf{q} = \mathbf{AB} - \mathbf{p}$ .
The desired distance is the length of $\mathbf{p}$ .
The desired distance is the length of $\mathbf{w}$ .
Find the vector $\mathbf{p} = \mathrm{proj}_{\mathbf{n}} \mathbf{AB}$ .
The desired distance is the length of $\mathbf{q}$ .
Find the vector $\mathbf{p} = \mathrm{proj}_{\mathbf{v}} \mathbf{AB}$ .
Find the vector $\mathbf{n} = \mathbf{v} \times \mathbf{w}$ .

Your solution: Put the statements in order in this column and press the Submit Answers button.

You can earn partial credit on this problem.