# Distance between a point and a line

Suppose you are given a point $B$ not on the line $\mathcal{L}: \ \mathbf{x}(t) = A + t \mathbf{v}$. What is a procedure you could use to find the distance between the point $B$ and the line $\mathcal{L}$? 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.
1. Find the vector $\mathbf{p} = \mathrm{proj}_{\mathbf{v}} \mathbf{AB}$.
2. Find the vector $\mathbf{p} = \mathrm{proj}_{\mathbf{AB}} \mathbf{v}$.
3. Construct the vector $\mathbf{AB}$.
4. The desired distance is the length of $\mathbf{v}$.
5. The desired distance is the length of $\mathbf{p}$.
6. The desired distance is the length of $\mathbf{q}$.
7. Find the vector $\mathbf{q} = \mathbf{AB} - \mathbf{p}$.
Your solution: Put the statements in order in this column and press the Submit Answers button.

# Distance between a point and a plane

Suppose you are given a point $B$ not on the plane $\mathcal{P} : \mathbf{n} \cdot ( \mathbf{x} - A) = 0$. What is a procedure you could use to find the distance from the point $B$ to the plane $\mathcal{P}$? Hint: draw a picture.
Statements to choose from: Drag these statements to the right column.
1. Find the vector $\mathbf{q} = \mathbf{AB} - \mathbf{p}$.
2. The desired distance is the length of $\mathbf{p}$.
3. Find the vector $\mathbf{w} = \mathbf{n} \times \mathbf{v}$.
4. Find the vector $\mathbf{p} = \mathrm{proj}_{\mathbf{n}} \mathbf{AB}$.
5. The desired distance is the length of $\mathbf{q}$.
6. The desired distance is the length of $\mathbf{w}$.
7. Construct the vector $\mathbf{AB}$.
Your solution: Put the statements in order in this column and press the Submit Answers button.