Problem 82
In contrast to the proof in Exercise \(81,\) we now use coordinates and position vectors to prove the same result. Without loss of generality, let \(P\left(x_{1}, y_{1}, 0\right)\) and \(Q\left(x_{2}, y_{2}, 0\right)\) be two points in the \(x y\) -plane and let \(R\left(x_{3}, y_{3}, z_{3}\right)\) be a third point, such that \(P, Q,\) and \(R\) do not lie on a line. Consider \(\triangle P Q R\) a. Let \(M_{1}\) be the midpoint of the side \(P Q\). Find the coordinates of \(M_{1}\) and the components of the vector \(\overrightarrow{R M}\) b. Find the vector \(\overrightarrow{O Z}_{1}\) from the origin to the point \(Z_{1}\) two-thirds of the way along \(\overrightarrow{R M}_{1}\). c. Repeat the calculation of part (b) with the midpoint \(M_{2}\) of \(R Q\) and the vector \(\overrightarrow{P M}_{2}\) to obtain the vector \(\overrightarrow{O Z}_{2}\) d. Repeat the calculation of part (b) with the midpoint \(M_{3}\) of \(P R\) and the vector \(\overline{Q M}_{3}\) to obtain the vector \(\overrightarrow{O Z}_{3}\) e. Conclude that the medians of \(\triangle P Q R\) intersect at a point. Give the coordinates of the point. f. With \(P(2,4,0), Q(4,1,0),\) and \(R(6,3,4),\) find the point at which the medians of \(\triangle P Q R\) intersect.
Problem 82
Prove the following vector properties using components. Then make a sketch to illustrate the property geometrically. Suppose \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\) are vectors in the \(x y\) -plane and a and \(c\) are scalars. $$(\mathbf{u}+\mathbf{v})+\mathbf{w}=\mathbf{u}+(\mathbf{v}+\mathbf{w})$$
Problem 83
Direction angles and cosines Let \(\mathbf{v}=\langle a, b, c\rangle\) and let \(\alpha, \beta\) and \(\gamma\) be the angles between \(\mathbf{v}\) and the positive \(x\) -axis, the positive \(y\) -axis, and the positive \(z\) -axis, respectively (see figure). a. Prove that \(\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=1\) b. Find a vector that makes a \(45^{\circ}\) angle with \(\mathbf{i}\) and \(\mathbf{j}\). What angle does it make with \(\mathbf{k} ?\) c. Find a vector that makes a \(60^{\circ}\) angle with i and \(\mathbf{j}\). What angle does it make with k? d. Is there a vector that makes a \(30^{\circ}\) angle with \(\mathbf{i}\) and \(\mathbf{j} ?\) Explain. e. Find a vector \(\mathbf{v}\) such that \(\alpha=\beta=\gamma .\) What is the angle?
Problem 83
The points \(P, Q, R,\) and \(S,\) joined by the vectors \(\mathbf{u}, \mathbf{v}, \mathbf{w},\) and \(\mathbf{x},\) are the vertices of a quadrilateral in \(\mathbb{R}^{3}\). The four points needn't lie in \(a\) plane (see figure). Use the following steps to prove that the line segments joining the midpoints of the sides of the quadrilateral form a parallelogram. The proof does not use a coordinate system. a. Use vector addition to show that \(\mathbf{u}+\mathbf{v}=\mathbf{w}+\mathbf{x}\) b. Let \(\mathbf{m}\) be the vector that joins the midpoints of \(P Q\) and \(Q R\) Show that \(\mathbf{m}=(\mathbf{u}+\mathbf{v}) / 2\) c. Let \(n\) be the vector that joins the midpoints of \(P S\) and \(S R\). Show that \(\mathbf{n}=(\mathbf{x}+\mathbf{w}) / 2\) d. Combine parts (a), (b), and (c) to conclude that \(\mathbf{m}=\mathbf{n}\) e. Explain why part (d) implies that the line segments joining the midpoints of the sides of the quadrilateral form a parallelogram.
Problem 83
An object moves along a path given by $$\mathbf{r}(t)=\langle a \cos t+b \sin t, c \cos t+d \sin t, e \cos t+f \sin t\rangle$$ for \(0 \leq t \leq 2 \pi\) a. Show that the curve described by \(\mathbf{r}\) lies in a plane. b. What conditions on \(a, b, c, d, e,\) and \(f\) guarantee that the curve described by \(\mathbf{r}\) is a circle?
Problem 83
Prove the following vector properties using components. Then make a sketch to illustrate the property geometrically. Suppose \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\) are vectors in the \(x y\) -plane and a and \(c\) are scalars. $$a(c \mathbf{v})=(a c) \mathbf{v}$$
Problem 84
Prove the following vector properties using components. Then make a sketch to illustrate the property geometrically. Suppose \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\) are vectors in the \(x y\) -plane and a and \(c\) are scalars. $$a(\mathbf{u}+\mathbf{v})=a \mathbf{u}+a \mathbf{v}$$
Problem 84
Show that the formula defining the torsion, \(\tau=-\frac{d \mathbf{B}}{d s} \cdot \mathbf{N},\) is equivalent to \(\tau=-\frac{1}{|\mathbf{v}|} \frac{d \mathbf{B}}{d t} \cdot \mathbf{N} .\) The second formula is generally easier to use.
Problem 84
Cauchy-Schwarz Inequality The definition \(\mathbf{u} \cdot \mathbf{v}=|\mathbf{u}||\mathbf{v}| \cos \theta\) implies that \(|\mathbf{u} \cdot \mathbf{v}| \leq|\mathbf{u}||\mathbf{v}|\) (because \(|\cos \theta| \leq 1\) ). This inequality, known as the Cauchy-Schwarz Inequality, holds in any number of dimensions and has many consequences. What conditions on \(\mathbf{u}\) and \(\mathbf{v}\) lead to equality in the CauchySchwarz Inequality?
Problem 84
Derivative rules Suppose \(\mathbf{u}\) and \(\mathbf{v}\) are differentiable functions at \(t=0\) with \(\mathbf{u}(0)=\langle 0,1,1\rangle, \mathbf{u}^{\prime}(0)=\langle 0,7,1\rangle\) \(\mathbf{v}(0)=\langle 0,1,1\rangle,\) and \(\mathbf{v}^{\prime}(0)=\langle 1,1,2\rangle .\) Evaluate the following expressions. a. \(\left.\frac{d}{d t}(\mathbf{u} \cdot \mathbf{v})\right|_{t=0}\) b. \(\left.\frac{d}{d t}(\mathbf{u} \times \mathbf{v})\right|_{t=0}\) c. \(\left.\frac{d}{d t}(\cos t \mathbf{u}(t))\right|_{t=0}\)
