Distance between parallel lines

Suppose you are given two parallel lines $\mathcal{L}_1: \ \mathbf{x_1}(s) = A + s \mathbf{v}$ and $\mathcal{L}_2: \ \mathbf{x_2}(t) = B + t \mathbf{w}$ . What is a procedure you could use to find the distance between these two parallel lines? 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 $\mathbf{p} = \mathrm{proj}_{\mathbf{v}} \mathbf{AB}$ .
Find the vector $\mathbf{p} = \mathrm{proj}_{\mathbf{n}} \mathbf{AB}$ .
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 $\mathbf{n}$ .
The desired distance is the length of $\mathbf{q}$ .
Construct the vector $\mathbf{AB}$ .
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 $\mathbf{p} = \mathrm{proj}_{\mathbf{v}} \mathbf{AB}$ .
Find the vector $\mathbf{p} = \mathrm{proj}_{\mathbf{n}} \mathbf{AB}$ .
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 $\mathbf{n}$ .
The desired distance is the length of $\mathbf{q}$ .
Construct the vector $\mathbf{AB}$ .
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.

Distance between parallel planes

Suppose you are given two parallel planes $\mathcal{P}_1 : n_1 (x - a_1) + n_2 (y - a_2) + n_3 (z - a_3) = 0$ and $\mathcal{P}_2 : m_1 (x - b_1) + m_2 (y - b_2) + m_3 (z - b_3) = 0$ . What is a procedure you could use to find the distance between these two parallel planes? Hint: draw a picture.

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

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

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

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

You can earn partial credit on this problem.