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--- lecture_23 [2017/04/25 09:22] – [Distance between lines in $\mathbb{R}^3$ in general] rupert
+++ lecture_23 [2017/05/06 09:59] (current) – rupert
@@ Line 1: / Line 1: @@
 ===== The distance from a point to a line =====
-==== Dot product method ====
+==== Cross product method ====
 Suppose $L$ is a line in $\def\rt{\mathbb{R}^3}\def\rn{\mathbb{R}^n}\rt$. Let $A$ be a point on $L$ and let $\def\vv{\vec v}\vv$ be a direction vector along $L$.
@@ Line 22: / Line 22: @@
 \[ \def\dist{\text{dist}}\dist(B,L)=\frac{\|\vec{AB}\times\vv\|}{\|\vv\|}=\frac{2\sqrt{7^2+8^2+4^2}}{\sqrt{4^2+1^2+5^2}} = \frac{2\sqrt{129}}{\sqrt{42}} \approx 3.5051.\]
-==== Cross product method ====
+==== Dot product method ====
 The method above relies on the cross product, so only works in $\def\c#1#2#3{\begin{bmatrix}#1\\#2\\#3\end{bmatrix}}\def\rt{\mathbb{R}^3}\def\rn{\mathbb{R}^n}\def\vv{\vec v}\def\dist{\text{dist}}\rt$. The following alternative method works in $\rn$ for any $n$.