91Ó°ÊÓ

Use vector methods to show that the perpendicular bisectors of the sides of a triangle intersect in a point, as follows. Assume the triangle \(O A B\) has one vertex at the origin, and let \(\mathbf{x}=\overrightarrow{O A}\) and \(\mathbf{y}=\overrightarrow{O B}\). Let \(\mathbf{z}\) be the point of intersection of the perpendicular bisectors of \(\overline{O A}\) and \(\overline{O B}\). Show that \(\mathrm{z}\) lies on the perpendicular bisector of \(\overline{A B}\). (Hint: What is the dot product of \(\mathbf{z}-\frac{1}{2}(\mathbf{x}+\mathbf{y})\) with \(\mathbf{x}-\mathbf{y}\) ?)

Verify algebraically that the following properties of vector arithmetic hold. (Do so for \(n=2\) if the general case is too intimidating.) Give the geometric interpretation of each property. a. For all \(\mathbf{x}, \mathbf{y} \in \mathbb{R}^{n}, \mathbf{x}+\mathbf{y}=\mathbf{y}+\mathbf{x}\). b. For all \(\mathbf{x}, \mathbf{y}, \mathbf{z} \in \mathbb{R}^{n},(\mathbf{x}+\mathbf{y})+\mathbf{z}=\mathbf{x}+(\mathbf{y}+\mathbf{z})\). c. \(\mathbf{0}+\mathbf{x}=\mathbf{x}\) for all \(\mathbf{x} \in \mathbb{R}^{n}\). d. For each \(\mathbf{x} \in \mathbb{R}^{n}\), there is a vector \(-\mathbf{x}\) so that \(\mathbf{x}+(-\mathbf{x})=\mathbf{0}\). e. For all \(c, d \in \mathbb{R}\) and \(\mathbf{x} \in \mathbb{R}^{n}, c(d \mathbf{x})=(c d) \mathbf{x}\). f. For all \(c \in \mathbb{R}\) and \(\mathbf{x}, \mathbf{y} \in \mathbb{R}^{n}, c(\mathbf{x}+\mathbf{y})=c \mathbf{x}+c \mathbf{y}\). g. For all \(c, d \in \mathbb{R}\) and \(\mathbf{x} \in \mathbb{R}^{n},(c+d) \mathbf{x}=c \mathbf{x}+d \mathbf{x}\). h. For all \(\mathbf{x} \in \mathbb{R}^{n}, 1 \mathbf{x}=\mathbf{x}\).