Chapter 1

Find the transformation matrix that rotates the axis \(x_{3}\) of a rectangular coordinate system \(45^{\circ}\) toward \(x_{1}\) around the \(x_{2}\) -axis.

Find the transformation matrix that rotates a rectangular coordinate system through an angle of \(120^{\circ}\) about an axis making equal angles with the original three coordinate axes.

Show (a) \((\mathbf{A B})^{t}=\mathbf{B}^{t} \mathbf{A}^{t}\) (b) \((A B)^{-1}=B^{-1} A^{-1}\)

Show by direct expansion that \(|\boldsymbol{\lambda}|^{2}=1 .\) For simplicity, take \(\boldsymbol{\lambda}\) to be a twodimensional orthogonal transformation matrix.

Consider a unit cube with one corner at the origin and three adjacent sides lying along the three axes of a rectangular coordinate system. Find the vectors describing the diagonals of the cube. What is the angle between any pair of diagonals?

Let \(\mathbf{A}\) be a vector from the origin to a point \(P\) fixed in space. Let \(\mathbf{r}\) be a vector from the origin to a variable point \(Q\left(x_{1}, x_{2}, x_{3}\right) .\) Show that $$\mathbf{A} \cdot \mathbf{r}=A^{2}$$ is the equation of a plane perpendicular to A and passing through the point \(P\)

For the two vectors $$\mathbf{A}=\mathbf{i}+2 \mathbf{j}-\mathbf{k}, \quad \mathbf{B}=-2 \mathbf{i}+3 \mathbf{j}+\mathbf{k}$$ find (a) \(\mathbf{A}-\mathbf{B}\) and \(|\mathbf{A}-\mathbf{B}|\) (b) component of B along A (c) angle between A and B (d) \(\mathbf{A} \times \mathbf{B}\) (e) \((\mathbf{A}-\mathbf{B}) \times(\mathbf{A}+\mathbf{B})\)

A particle moves in a plane elliptical orbit described by the position vector $$\mathbf{r}=2 b \sin \omega t \mathbf{i}+b \cos \omega t \mathbf{j}$$ (a) Find \(\mathbf{v}, \mathbf{a},\) and the particle speed. (b) What is the angle between \(\mathbf{v}\) and a at time \(t=\pi / 2 \omega ?\)

Show that the triple scalar product \((\mathbf{A} \times \mathbf{B}) \cdot \mathbf{C}\) can be written as $$(\mathbf{A} \times \mathbf{B}) \cdot \mathbf{C}=\left|\begin{array}{lll} A_{1} & A_{2} & A_{3} \\ B_{1} & B_{2} & B_{3} \\ C_{1} & C_{2} & C_{3} \end{array}\right|$$ Show also that the product is unaffected by an interchange of the scalar and vector product operations or by a change in the order of \(\mathbf{A}, \mathbf{B}, \mathbf{C},\) as long as they are in cyclic order; that is, $$(\mathbf{A} \times \mathbf{B}) \cdot \mathbf{C}=\mathbf{A} \cdot(\mathbf{B} \times \mathbf{C})=\mathbf{B} \cdot(\mathbf{C} \times \mathbf{A})=(\mathbf{C} \times \mathbf{A}) \cdot \mathbf{B}, \quad \text { etc. }$$ We may therefore use the notation ABC to denote the triple scalar product. Finally, give a geometric interpretation of \(\mathbf{A B C}\) by computing the volume of the parallelepiped defined by the three vectors \(\mathbf{A}, \mathbf{B}, \mathbf{C}\)

Let \(\mathbf{a}, \mathbf{b}, \mathbf{c}\) be three constant vectors drawn from the origin to the points \(A, B, C\) What is the distance from the origin to the plane defined by the points \(A, B, C\) ? What is the area of the triangle \(A B C\) ?

\(\mathbf{X}\) is an unknown vector satisfying the following relations involving the known vectors \(\mathbf{A}\) and \(\mathbf{B}\) and the scalar \(\phi\) $$\mathbf{A} \times \mathbf{X}=\mathbf{B}, \quad \mathbf{A} \cdot \mathbf{X}=\phi$$ Express \(\mathbf{X}\) in terms of \(\mathbf{A}, \mathbf{B}, \phi,\) and the magnitude of \(\mathbf{A}\)

Consider the following matrices: $$\mathbf{A}=\left(\begin{array}{rrr} 1 & 2 & -1 \\ 0 & 3 & 1 \\ 2 & 0 & 1 \end{array}\right), \quad \mathbf{B}=\left(\begin{array}{rrr} 2 & 1 & 0 \\ 0 & -1 & 2 \\ 1 & 1 & 3 \end{array}\right), \quad \mathbf{C}=\left(\begin{array}{rr} 2 & 1 \\ 4 & 3 \\ 1 & 0 \end{array}\right)$$ Find the following (a) \(|\mathbf{A B}|\) (b) \(\mathbf{A C}\) (c) ABC (d) \(\mathbf{A B}-\mathbf{B}^{t \mathbf{A}^{t}}\)

Find the values of \(\alpha\) needed to make the following transformation orthogonal. $$ \left(\begin{array}{rrr} 1 & 0 & 0 \\ 0 & \alpha & -\alpha \\ 0 & \alpha & \alpha \end{array}\right) $$

What surface is represented by \(\mathbf{r} \cdot \mathbf{a}=\) const. that is described if a is a vector of constant magnitude and direction from the origin and \(\mathbf{r}\) is the position vector to the point \(P\left(x_{1}, x_{2}, x_{3}\right)\) on the surface?

Obtain the cosine law of plane trigonometry by interpreting the product \((\mathbf{A}-\mathbf{B})\) \((\mathbf{A}-\mathbf{B})\) and the expansion of the product.

Obtain the sine law of plane trigonometry by interpreting the product \(\mathbf{A} \times \mathbf{B}\) and the alternate representation \((\mathbf{A}-\mathbf{B}) \times \mathbf{B}\).

Derive the following expressions by using vector algebra: (a) \(\cos (\alpha-\beta)=\cos \alpha \cos \beta+\sin \alpha \sin \beta\) (b) \(\sin (\alpha-\beta)=\sin \alpha \cos \beta-\cos \alpha \sin \beta\)

Show that (a) \(\sum_{i, j} \varepsilon_{i j k} \delta_{i j}=0\) (b) \(\sum_{j, k} \varepsilon_{i j k} \varepsilon_{l j k}=2 \delta_{i l}\) (c) \(\sum_{i, j, k} \varepsilon_{i j k} \varepsilon_{i j k}=6\)

Show (see also Problem \(1-11\) ) that $$\mathbf{A B C}=\sum_{i, j, k} \varepsilon_{i j k} A_{i} B_{j} C_{k}$$

Evaluate the \(\operatorname{sum} \sum_{k} \varepsilon_{i j k} \varepsilon_{l m k}(\text { which contains } 3\) terms) by considering the result for all possible combinations of \(i, j, l, m ;\) that is, (a) \(i=j\) (b) \(i=l\) (c) \(i=m\) (d) \(j=l\) (e) \(j=m\) (f) \(l=m\) (g) \(i \neq l\) or \(m\) (h) \(j \neq l\) or \(m\) Show that $$\sum_{k} \varepsilon_{i j k} \varepsilon_{l m k}=\delta_{i l} \delta_{j m}-\delta_{i m} \delta_{j l}$$ and then use this result to prove $$\mathbf{A} \times(\mathbf{B} \times \mathbf{C})=(\mathbf{A} \cdot \mathbf{C}) \mathbf{B}-(\mathbf{A} \cdot \mathbf{B}) \mathbf{C}$$

Use the \(\varepsilon_{\text {ijk }}\) notation and derive the identity $$(\mathbf{A} \times \mathbf{B}) \times(\mathbf{C} \times \mathbf{D})=(\mathbf{A B D}) \mathbf{C}-(\mathbf{A B C}) \mathbf{D}$$

Let \(\mathbf{A}\) be an arbitrary vector, and let e be a unit vector in some fixed direction. Show that $$\mathbf{A}=\mathbf{e}(\mathbf{A} \cdot \mathbf{e})+\mathbf{e} \times(\mathbf{A} \times \mathbf{e})$$ What is the geometrical significance of each of the two terms of the expansion?

A particle moves with \(v=\) const. along the curve \(r=k(1+\cos \theta)\) (a cardioid). Find \(\ddot{\mathbf{r}} \cdot \mathbf{e}_{r}=\mathbf{a} \cdot \mathbf{e}_{r},|\mathbf{a}|,\) and \(\dot{\theta}\).

If \(\mathbf{r}\) and \(\mathbf{r}=\mathbf{v}\) are both explicit functions of time, show that $$\frac{d}{d t}[\mathbf{r} \times(\mathbf{v} \times \mathbf{r})]=r^{2} \mathbf{a}+(\mathbf{r} \cdot \mathbf{v}) \mathbf{v}-\left(v^{2}+\mathbf{r} \cdot \mathbf{a}\right) \mathbf{r}$$

Show that $$\nabla(\ln |\mathbf{r}|)=\frac{\mathbf{r}}{r^{2}}$$

Find the angle between the surfaces defined by \(r^{2}=9\) and \(x+y+z^{2}=1\) at the point (2,-2,1).

Show that \(\nabla(\phi \psi)=\phi \nabla \psi+\psi \nabla \phi\).

Show that (a) \(\nabla r^{n}=n r^{(n-2)} \mathbf{r}\) (b) \(\nabla f(r)=\frac{\mathbf{r}}{r} \frac{d f}{d r}\) (c) \(\nabla^{2}(\ln r)=\frac{1}{r^{2}}\)

Show that $$\int(2 a \mathbf{r} \cdot \mathbf{i}+2 b \mathbf{r} \cdot \mathbf{r}) d t=a \mathbf{r}^{2}+b \dot{r}^{2}+\text { const }$$ where \(\mathbf{r}\) is the vector from the origin to the point \(\left(x_{1}, x_{2}, x_{3}\right) .\) The quantities \(r\) and \(\dot{r}\) are the magnitudes of the vectors \(\mathbf{r}\) and \(\mathbf{r}\), respectively, and \(a\) and \(b\) are constants.

Show that $$\int\left(\frac{\dot{\mathbf{r}}}{r}-\frac{\mathbf{r} \dot{r}}{r^{2}}\right) d t=\frac{\mathbf{r}}{r}+\mathbf{C}$$ where \(\mathbf{C}\) is a constant vector.

Show that the volume common to the intersecting cylinders defined by \(x^{2}+y^{2}=a^{2}\) and \(x^{2}+z^{2}=a^{2}\) is \(V=16 a^{3} / 3\).

Find the value of the integral \(\int_{s} \mathbf{A} \cdot d \mathbf{a},\) where \(\mathbf{A}=\left(x^{2}+y^{2}+z^{2}\right)(x \mathbf{i}+y \mathbf{j}+z \mathbf{k})\) and the surface \(S\) is defined by the sphere \(R^{2}=x^{2}+y^{2}+z^{2} .\) Do the integral directly and also by using Gauss's theorem.

A plane passes through the three points \((x, y, z)=(1,0,0),(0,2,0),(0,0,3)\) (a) Find a unit vector perpendicular to the plane. (b) Find the distance from the point (1,1,1) to the closest point of the plane and the coordinates of the closest point.

The height of a hill in meters is given by \(z=2 x y-3 x^{2}-4 y^{2}-18 x+28 y+12\) where \(x\) is the distance east and \(y\) is the distance north of the origin. (a) Where is the top of the hill and how high is it? (b) How steep is the hill at \(x=y=1\), that is, what is the angle between a vector perpendicular to the hill and the \(z\) axis? \((c)\) In which compass direction is the slope at \(x=y=1\) steepest?

For what values of \(a\) are the vectors \(\mathbf{A}=2 a \mathbf{i}-2 \mathbf{j}+a \mathbf{k}\) and \(\mathbf{B}=a \mathbf{i}+2 a \mathbf{j}+2 \mathbf{k}\) perpendicular?