Chapter 4

$$ \text { Solve the equations } \mathbf{x}+(\mathbf{x} \cdot \mathbf{i}) \mathbf{i}=\mathbf{j} . \text { and } \mathbf{x}+(\mathbf{x} \times \mathbf{i})=\mathbf{j} $$

Show that the distance between the point \((1,1,1)\) and the line through \((2,0,3)\) and \((-1,0,1)\) is \(\sqrt{29 / 13}\).

Show that the determinants $$ \left|\begin{array}{ccc} 1 & 10 & 4 \\ 8 & 82 & 30 \\ 6 & 62 & 23 \end{array}\right|, \quad\left|\begin{array}{ccc} 100 & 20 & 13 \\ 6 & 1 & 2 \\ 80 & 20 & 5 \end{array}\right|, \quad\left|\begin{array}{lll} 999 & 998 & 997 \\ 996 & 995 & 994 \\ 993 & 992 & 991 \end{array}\right| $$ are \(2,-380\) and 0 , respectively.

$$ \text { Prove that }(\mathbf{a} \times \mathbf{b}) \times \mathbf{c}=\mathbf{a} \times(\mathbf{b} \times \mathbf{c}) \text { if and only if }(\mathbf{a} \times \mathbf{c}) \times \mathbf{b}=\mathbf{0} \text {. } $$

Suppose that \(\mathbf{a} \neq \mathbf{0}\). Show that \(\mathbf{x}=\mathbf{y}\) if and only if \(\mathbf{a} \cdot \mathbf{x}=\mathbf{a} \cdot \mathbf{y}\) and \(\mathbf{a} \times \mathbf{x}=\mathbf{a} \times \mathbf{y}\)

Let \(\theta\) be the acute angle formed by two diagonals of a cube. Show that \(\cos \theta=1 / 3\), and hence find \(\theta\)

Show that the distance between the point \((-3,0,1)\) and the line given by \((1,0,2)+t(1,1,2)\), where \(t \in \mathbb{R}\), is \(7 / \sqrt{3}\)

Evaluate the determinants $$ \left|\begin{array}{ccc} 1 & 1 & 1 \\ x & a & b \\ x^{2} & a^{2} & b^{2} \end{array}\right|,\left|\begin{array}{ccc} x & a & b \\ x^{2} & a^{2} & b^{2} \\ x^{3} & a^{3} & b^{3} \end{array}\right| $$ and factorize both answers.

Prove the following vector identities: $$ \begin{aligned} &\mathbf{a} \times(\mathbf{b} \times \mathbf{c})+\mathbf{b} \times(\mathbf{c} \times \mathbf{a})+\mathbf{c} \times(\mathbf{a} \times \mathbf{b})=\mathbf{0} \\\ &(\mathbf{a} \times \mathbf{b}) \times(\mathbf{c} \times \mathbf{d})=[\mathbf{a}, \mathbf{b}, \mathbf{d}] \mathbf{c}-[\mathbf{a}, \mathbf{b}, \mathbf{c}] \mathbf{d} \\ &(\mathbf{a} \times \mathbf{b}) \cdot((\mathbf{c} \times \mathbf{d}) \times(\mathbf{e} \times \mathbf{f}))=[\mathbf{a}, \mathbf{b}, \mathbf{d}][\mathbf{c}, \mathbf{e}, \mathbf{f}]-[\mathbf{a}, \mathbf{b}, \mathbf{c}][\mathbf{d}, \mathbf{e}, \mathbf{f}] \\ &(\mathbf{b} \times \mathbf{c}) \cdot(\mathbf{a} \times \mathbf{d})+(\mathbf{c} \times \mathbf{a}) \cdot(\mathbf{b} \times \mathbf{d})+(\mathbf{a} \times \mathbf{b}) \cdot(\mathbf{c} \times \mathbf{d})=0 \end{aligned} $$

Use the vector product to find the area of the triangle with vertices \((1,2,0)\), \((2,5,2)\) and \((4,-1,2)\).

Let \(Q\) be a quadilateral whose sides have lengths \(\ell_{1}, \ell_{2}, \ell_{3}\) and \(\ell_{4}\) taken in this order around the quadilateral. Show that the diagonals of \(Q\) are orthogonal if and only if \(\ell_{1}^{2}+\ell_{3}^{2}=\ell_{2}^{2}+\ell_{4}^{2}\). [Hint: let the vertices of \(Q\) be \(\mathbf{a}, \mathbf{b}, \mathbf{c}\) and \(\mathbf{d}\).] Deduce that the diagonals of a rectangle are orthogonal if and only if the rectangle is a square, and that the diagonals of a parallelogram are orthogonal if and only if the parallelogram is a rhombus.

Given vectors \(\mathbf{a}\) and \(\mathbf{b}\), and positive numbers \(\ell_{1}\) and \(\ell_{2}\) such that \(\ell_{1}+\ell_{2}=\) \(\|\mathbf{a}-\mathbf{b}\|\), let \(\mathbf{c}\) be the unique vector on \([\mathbf{a}, \mathbf{b}]\) such that \(\|\mathbf{c}-\mathbf{a}\|=\ell_{1}\) and \(\|\mathbf{c}-\mathbf{b}\|=\ell_{2}\). By writing \(\mathbf{c}-\mathbf{a}=t(\mathbf{b}-\mathbf{a})\), for some real \(t\), show that $$ \mathbf{c}=\frac{\ell_{2}}{\ell_{1}+\ell_{2}} \mathbf{a}+\frac{\ell_{1}}{\ell_{1}+\ell_{2}} \mathbf{b} $$ What is the mid-point of the segment \([\mathbf{a}, \mathbf{b}] ?\)

Show that the solution of the simultaneous equations \(\mathbf{x}+(\mathbf{c} \times \mathbf{y})=\mathbf{a}\) and \(y+(\mathbf{c} \times \mathbf{x})=\mathbf{b}\) is given by $$ \begin{aligned} &\mathbf{x}=[(\mathbf{a} \cdot \mathbf{c}) \mathbf{c}+\mathbf{a}+\mathbf{b} \times \mathbf{c}] /\left(1+\|\mathbf{c}\|^{2}\right) \\ &\mathbf{y}=[(\mathbf{b} \cdot \mathbf{c}) \mathbf{c}+\mathbf{b}+\mathbf{a} \times \mathbf{c}] /\left(1+\|\mathbf{c}\|^{2}\right) \end{aligned} $$

Show that the distance between the two lines in the direction \((1,2,1)\) that pass through the points \((4,2,-1)\) and \((3,1,0)\), respectively is \(7 / \sqrt{21}\).

Show that for any vectors \(\mathbf{a}, \mathbf{b}, \mathbf{c}, \mathbf{u}, \mathbf{v}, \mathbf{w}\) in \(\mathbb{R}^{3}\), $$ [\mathbf{a}, \mathbf{b}, \mathbf{c}][\mathbf{u}, \mathbf{v}, \mathbf{w}]=\left|\begin{array}{lll} \mathbf{a} \cdot \mathbf{u} & \mathbf{a} \cdot \mathbf{v} & \mathbf{a} \cdot \mathbf{w} \\ \mathbf{b} \cdot \mathbf{u} & \mathbf{b} \cdot \mathbf{v} & \mathbf{b} \cdot \mathbf{w} \\ \mathbf{c} \cdot \mathbf{u} & \mathbf{c} \cdot \mathbf{v} & \mathbf{c} \cdot \mathbf{w} \end{array}\right| $$

$$ \text { Prove that }(\mathbf{a} \times \mathbf{b}) \cdot(\mathbf{c} \times \mathbf{d})=(\mathbf{a} \cdot \mathbf{c})(\mathbf{b} \cdot \mathbf{d})-(\mathbf{a} \cdot \mathbf{d})(\mathbf{b} \cdot \mathbf{c}) $$

Suppose that \(\mathbf{u}=s_{1} \mathbf{i}+s_{2} \mathbf{j}\) and \(\mathbf{v}=t_{1} \mathbf{i}+t_{2} \mathbf{j}\), where \(s_{1}, s_{2}, t_{1}\) and \(t_{2}\) are real numbers. Find a necessary and sufficient condition on these real numbers such that every vector in the plane of \(\mathbf{i}\) and \(\mathbf{j}\) can be expressed as a linear combination of the vectors \(\mathbf{u}\) and \(\mathbf{v}\).

Show that the distance between the two skew lines given in parametric form by \((1,2,3)+t(2,0,1)\) and \((0,0,1)+t(1,0,1)\) is 2 .

Suppose that a, b and \(c\) do not lie on any straight line. Show that the normal to the plane that contains them lies in the direction \((\mathbf{a} \times \mathbf{b})+\) \((\mathbf{b} \times \mathbf{c})+(\mathbf{c} \times \mathbf{a})\). Consider the special case \(\mathbf{c}=\mathbf{0}\).

Find the equation of the plane through the points \((0,1,2),(-4,3,1)\) and \((10,0,7)\)

Suppose that \(\mathbf{a} \times \mathbf{x} \neq \mathbf{0}\). Define vectors \(\mathbf{x}_{0}, \mathbf{x}_{1}, \ldots\) by \(\mathbf{x}_{0}=\mathbf{x}\) and \(\mathbf{x}_{n+1}=\mathbf{a} \times \mathbf{x}_{n} . \mathbf{A s}\left\|\mathbf{x}_{n}\right\| \leq\|\mathbf{a}\|^{n}\|\mathbf{x}\|\), it is clear that \(\mathbf{x}_{n} \rightarrow \mathbf{0}\) if \(\|\mathbf{a}\|<1\) What happens as \(n \rightarrow \infty\) if \(\|\mathbf{a}\|=1\), or if \(\|\mathbf{a}\|>1 ?\)

Find the intersection of the three planes given by \(\mathbf{x} \cdot \mathbf{a}=1, \mathbf{x} \cdot \mathbf{b}=2\) and \(\mathbf{x} \cdot \mathbf{c}=3\), where \(\mathbf{a}=(3,1,1), \mathbf{b}=(2,0,8)\) and \(\mathbf{c}=(1,0,2)\)

Find the vectorial equation of the line of intersection of the planes \(3 x_{1}+2 x_{2}+x_{3}=3\) and \(x_{1}+x_{2}+x_{3}=4\).

Consider the cube with vertices at the points \((r, s, t)\), where each of \(r, s\) and \(t\) is 0 or 1 . What is the surface area of the tetrahedron whose vertices are at the points \(\mathbf{0}\) and the centres of the three faces of the cube that do not contain 0 ?

Use vectors to show that the diagonals of a parallelogram bisect each other, and that the diagonals are orthogonal if and only if the parallelogram is a rhombus. Show that the midpoints of the sides of any auadrilateral form the vertices of a parallelogram

Let \(T\) be the tetrahedron with vertices \(\mathbf{0}, a \mathbf{i}, b \mathbf{j}\) and \(c \mathbf{k}\), and let the faces opposite these vertices have areas \(A_{0}, A_{a}, A_{b}\), and \(A_{c}\), respectively. Show that \(A_{0}^{2}=A_{a}^{2}+A_{b}^{2}+A_{c}^{2}\) (Pythagoras' theorem for a tetrahedron).

Show that the minimum distance between a pair of opposite edges of a regular tetrahedron \(T\) with edge length \(\ell\) is \(\ell / \sqrt{2}\).