4.6: Supplementary Exercises for Chapter 4
- Page ID
- 58853
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Supplementary Exercises for Chapter [chap:4]
solutions
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[supex:sup_ch4_ex1] Suppose that \(\mathbf{u}\) and \(\mathbf{v}\) are nonzero vectors. If \(\mathbf{u}\) and \(\mathbf{v}\) are not parallel, and \(a\mathbf{u} + b\mathbf{v} = a_{1}\mathbf{u} + b_{1}\mathbf{v}\), show that \(a = a_{1}\) and \(b = b_{1}\).
Consider a triangle with vertices \(A\), \(B\), and \(C\). Let \(E\) and \(F\) be the midpoints of sides \(AB\) and \(AC\), respectively, and let the medians \(EC\) and \(FB\) meet at \(O\). Write \(\longvect{EO} = s\longvect{EC}\) and \(\longvect{FO} = t\longvect{FB}\), where \(s\) and \(t\) are scalars. Show that \(s = t = \frac{1}{3}\) by expressing \(\longvect{AO}\) two ways in the form \(a\longvect{EO} + b\longvect{AC}\), and applying Exercise [supex:sup_ch4_ex1]. Conclude that the medians of a triangle meet at the point on each that is one-third of the way from the midpoint to the vertex (and so are concurrent).
A river flows at \(1\) km/h and a swimmer moves at \(2\) km/h (relative to the water). At what angle must he swim to go straight across? What is his resulting speed?
A wind is blowing from the south at \(75\) knots, and an airplane flies heading east at \(100\) knots. Find the resulting velocity of the airplane.
\(125\) knots in a direction \(\theta\) degrees east of north, where \(\cos\theta = 0.6\) (\(\theta = 53^\circ\) or \(0.93\) radians).
An airplane pilot flies at \(300\) km/h in a direction \(30^\circ\) south of east. The wind is blowing from the south at \(150\) km/h.
- Find the resulting direction and speed of the airplane.
- Find the speed of the airplane if the wind is from the west (at \(150\) km/h).
A rescue boat has a top speed of \(13\) knots. The captain wants to go due east as fast as possible in water with a current of \(5\) knots due south. Find the velocity vector \(\mathbf{v} = (x, y)\) that she must achieve, assuming the \(x\) and \(y\) axes point east and north, respectively, and find her resulting speed.
\((12, 5)\). Actual speed \(12\) knots.
A boat goes \(12\) knots heading north. The current is \(5\) knots from the west. In what direction does the boat actually move and at what speed?
Show that the distance from a point \(A\) (with vector \(\mathbf{a}\)) to the plane with vector equation \(\mathbf{n}\bullet \mathbf{p} = d\) is \(\frac{1}{\| n \|}|\mathbf{n}\bullet \mathbf{a} - d|\).
If two distinct points lie in a plane, show that the line through these points is contained in the plane.
The line through a vertex of a triangle, perpendicular to the opposite side, is called an altitude of the triangle. Show that the three altitudes of any triangle are concurrent. (The intersection of the altitudes is called the orthocentre of the triangle.) [Hint: If \(P\) is the intersection of two of the altitudes, show that the line through \(P\) and the remaining vertex is perpendicular to the remaining side.]