91Ó°ÊÓ

Given \(\triangle A B C\), let \(M\) and \(N\) be the midpoints of \(\overline{A B}\) and \(\overline{A C}\), respectively. Prove that \(\overrightarrow{M N}=\frac{1}{2} \overrightarrow{B C}\)

Use the algebraic properties of the dot product to show that $$ \|\mathbf{x}+\mathbf{y}\|^{2}+\|\mathbf{x}-\mathbf{y}\|^{2}=2\left(\|\mathbf{x}\|^{2}+\|\mathbf{y}\|^{2}\right) . $$ Interpret the result geometrically.

Use the dot product to prove the law of cosines: As shown in Figure 2.8, $$ c^{2}=a^{2}+b^{2}-2 a b \cos \theta . $$

a. Prove or give a counterexample: If \(A\) is an \(m \times n\) matrix and \(\mathbf{x} \in \mathbb{R}^{n}\) satisfies \(A \mathbf{x}=\mathbf{0}\), then either every entry of \(A\) is 0 or \(\mathbf{x}=\mathbf{0}\). b. Prove or give a counterexample: If \(A\) is an \(m \times n\) matrix and \(A \mathbf{x}=\mathbf{0}\) for every vector \(\mathbf{x} \in \mathbb{R}^{n}\), then every entry of \(A\) is 0 .

Use vector methods to prove that a triangle that is inscribed in a circle and has a diameter as one of its sides must be a right triangle. (Hint: See Figure 2.9. Express the vectors \(\mathbf{u}\) and \(\mathbf{v}\) in terms of \(\mathbf{x}\) and \(\mathbf{y}\).)

Suppose a living organism that can live to a maximum age of 3 years has Leslie matrix $$ A=\left[\begin{array}{ccc} 0 & 0 & 8 \\ \frac{1}{2} & 0 & 0 \\ 0 & \frac{1}{4} & 0 \end{array}\right] $$ Find a stable age distribution vector \(\mathbf{x}\), i.e., a vector \(\mathbf{x} \in \mathbb{R}^{3}\) with \(A \mathbf{x}=\mathbf{x}\).

Prove the triangle inequality: For any vectors \(\mathbf{x}, \mathbf{y} \in \mathbb{R}^{n}\), \(\|\mathbf{x}+\mathbf{y}\| \leq\|\mathbf{x}\|+\|\mathbf{y}\|\). (Hint: Use the dot product to calculate \(\|\mathbf{x}+\mathbf{y}\|^{2}\).)

Assume that \(\mathbf{u}\) and \(\mathbf{v}\) are parallel vectors in \(\mathbb{R}^{n}\). Prove that \(\operatorname{Span}(\mathbf{u}, \mathbf{v})\) is a line.

a. Let \(\mathbf{x}\) and \(\mathbf{y}\) be vectors with \(\|\mathbf{x}\|=\|\mathbf{y}\|\). Prove that the vector \(\mathbf{x}+\mathbf{y}\) bisects the angle between \(\mathbf{x}\) and \(\mathbf{y}\). (Hint: Because \(\mathbf{x}+\mathbf{y}\) lies in the plane spanned by \(\mathbf{x}\) and \(\mathbf{y}\), one has only to check that the angle between \(\mathbf{x}\) and \(\mathbf{x}+\mathbf{y}\) equals the angle between \(\mathbf{y}\) and \(\left.\mathbf{x}+\mathbf{y}_{-}\right)\) b. More generally, if \(\mathbf{x}\) and \(\mathbf{y}\) are arbitrary nonzero vectors, let \(a=\|\mathbf{x}\|\) and \(b=\|\mathbf{y}\|\). Prove that the vector \(b \mathbf{x}+a \mathbf{y}\) bisects the angle between \(\mathbf{x}\) and \(\mathbf{y}\).

Use vector methods to prove that the diagonals of a parallelogram bisect the vertex angles if and only if the parallelogram is a rhombus. (Hint: Use Exercise 21.)