(a) Express vector \(\mathbf{c}\) in the form \(\mathbf{c}=n\,\mathbf{a}+m\,\mathbf{b}\).
(b) Calculate \(|\mathbf{a}|\) and \(|\mathbf{b}|\).
(c) Calculate scalar product \(\mathbf{a}\cdot\mathbf{b}\).
(d) Find the angle between \(\mathbf{a}\) and \(\mathbf{b}\).
Solutions: (a) \(\mathbf{c}=3\mathbf{a}+5\mathbf{b}\); (b) \(|\mathbf{a}|=5\), \(|\mathbf{b}|=2\sqrt{2}\); (c) \(\mathbf{a}\cdot\mathbf{b}=-2\); (d) \(\theta\approx98.1^\circ\)(a) Find the coordinates of the vertex \(D\).
(b) Calculate angle \(\alpha=B\hat{A}D\).
(c) Calculate the length of the longer diagonal.
Solutions: (a) \(D(10,18)\); (b) \(\alpha\approx36.4^\circ\); (c) \(AC=29\)(a) Show that this quadrilateral is not a parallelogram.
(b) Show that sides \(AB\) and \(CD\) are parallel.
Straight line \(L\) passes through points \(A\) and \(B\).
(c) Write the equation of \(L\) in the form \(y=mx+c\).
Solutions: (a) \(\overrightarrow{AB}\ne\overrightarrow{DC}\), \(\overrightarrow{AD}\ne\overrightarrow{BC}\) ; (b) \(\overrightarrow{AB}=\frac{3}{2}\overrightarrow{DC}\); (c) \(y=\frac{1}{4}x+\frac{7}{4}\)(a) Calculate lengths of all three sides.
(b) Calculate the largest angle in this triangle.
(c) Calculate the area of this triangle.
Solutions: (a) \(a=BC=7,~ b=AC=11,~ c=AB=14\) ; (b) \(\gamma=A\hat{C}B\approx99.7^\circ\); (c) \(A\approx37.9\)(a) Find the coordinates of points \(B\) and \(C\).
(b) Write down \(\overrightarrow{BA}\).
(c) Calculate the angle \(\beta=A\hat{B}C\).
Point \(D\) is the midpoint of the side \(AC\).
(d) Calculate the distance \(DB\).
Solutions: (a) \(B(8,1),~ C(11,7)\); (b) \(\overrightarrow{BA}=\left(\begin{array}{c} -9 \\ 2 \end{array}\right)\); (c) \(\beta\approx104^\circ\); (d) \(DB=5\)(a) Write \(\overrightarrow{QR}\).
(b) Show that \(\overrightarrow{PQ}\) is perpendicular to \(\overrightarrow{QR}\).
Unit vector \(\mathbf{u}\) has the same direction as \(\overrightarrow{PQ}\).
(c) Write \(\mathbf{u}\) in terms of \(\mathbf{i},~ \mathbf{j}\) and \(\mathbf{k}\).
Solutions: (a) \(\overrightarrow{QR}=\left(\begin{array}{c} 2 \\ 6 \\ -5 \end{array}\right)=2\mathbf{i}+6\mathbf{j}-5\mathbf{k}\); (b) \(\overrightarrow{PQ}\cdot\overrightarrow{QR}=0\); (c) \(\mathbf{u}=\frac{2}{3}\mathbf{i}+\frac{1}{3}\mathbf{j}+\frac{2}{3}\mathbf{k}\)(a) Find \(p\), given that \(p\) is a positive number.
Consider two other vectors: \(\mathbf{c}=\mathbf{a}+\mathbf{b}\) and \(\mathbf{d}=\mathbf{a}-\mathbf{b}\).
(b) Write down \(\mathbf{c}\) and \(\mathbf{d}\).
(c) Show that \(\mathbf{c}\) is perpendicular to \(\mathbf{d}\).
Solutions: (a) \(p=5\); (b) \(\mathbf{c}=\left(\begin{array}{c} 15 \\ 24 \\ 7 \end{array}\right)\), \(\mathbf{d}=\left(\begin{array}{c} -5 \\ 4 \\ -3 \end{array}\right)\); (c) \(\mathbf{c}\cdot\mathbf{d}=0\)(a) Write down \(\overrightarrow{PQ}\).
(b) Write the equation of \(L\) in the form \(\mathbf{r}=\mathbf{a}+\mathbf{b}\,t\).
(c) Calculate the acute angle formed by \(L\) and the horizontal axis.
Solutions: (a) \(\overrightarrow{PQ}=\left(\begin{array}{c} 4 \\ 4 \end{array}\right)\); (b) \(\mathbf{r}=\left(\begin{array}{c} 5 \\ 1 \end{array}\right)+\left(\begin{array}{c} 4 \\ 4 \end{array}\right)t\); (c) \(\varphi=45^\circ\)(a) Find \(\overrightarrow{AB}\).
(b) Write the equation of \(L_1\) in the form \(\mathbf{r}=\mathbf{a}+\mathbf{b}\,t\).
Line \(L_2\) has the equation \(\mathbf{r}=\left(\begin{array}{c} 12 \\ 10 \\ 0 \end{array}\right)+\left(\begin{array}{c} 1 \\ 5 \\ 2 \end{array}\right)t\).
(c) Find the angle between \(L_1\) and \(L_2\).
(d) Find the intersection point.
Solutions: (a) \(\overrightarrow{AB}=\left(\begin{array}{c} 2 \\ 3 \\ 1 \end{array}\right)\); (b) \(\mathbf{r}=\left(\begin{array}{c} 4 \\ 5 \\ -1 \end{array}\right)+\left(\begin{array}{c} 2 \\ 3 \\ 1 \end{array}\right)t\); (c) \(\varphi\approx22.0^\circ\); (d) \(P(14,20,4)\)(a) Write the equation of \(L\) in the form \(\mathbf{r}=\mathbf{a}+\mathbf{b}\,t\).
(b) Write the equation of \(L\) in the form \(y=mx+c\).
(c) Write the equation of \(L\) in the form \(px+qy=s\), where \(p,~ q\) and \(s\) are integers.
Solutions: (a) \(\mathbf{r}=\left(\begin{array}{c} 4 \\ 2 \end{array}\right)+\left(\begin{array}{c} 4 \\ -3 \end{array}\right)t\); (b) \(y=-\frac{3}{4}x+5\); (c) \(3x+4y=20\)(a) Write the equation of \(L_1\) in the form \(\mathbf{r}=\mathbf{a}+\mathbf{b}\,t\).
(b) Show that \(L_1\) passes through the point \(A(11,2)\).
Line \(L_2\) passes through \(A\) and is perpendicular to \(L_1\).
(c) Write the equation of \(L_2\) in the form \(y=mx+c\).
Solutions: (a) \(\mathbf{r}=\left(\begin{array}{c} 2 \\ -1 \end{array}\right)+\left(\begin{array}{c} 3 \\ 1 \end{array}\right)t\); (c) \(y=-3x+35\)