# 12: Three Dimensions (Exercises)

These are homework exercises to accompany David Guichard's "General Calculus" Textmap. Complementary General calculus exercises can be found for other Textmaps and can be accessed here.

## 12.1: The Coordinate System

**Ex 12.1.1** Sketch the location of the points \((1,1,0)\), \((2,3,-1)\), and \((-1,2,3)\) on a single set of axes.

**Ex 12.1.2** Describe geometrically the set of points \((x,y,z)\) that satisfy \(z=4\).

**Ex 12.1.3** Describe geometrically the set of points \((x,y,z)\) that satisfy \(y=-3\).

**Ex 12.1.4** Describe geometrically the set of points \((x,y,z)\) that satisfy \(x+y=2\).

**Ex 12.1.5** The equation \(x+y+z=1\) describes some collection of points in \( \R^3\). Describe and sketch the points that satisfy \(x+y+z=1\) and are in the \(x\)-\)y\) plane, in the \(x\)-\)z\) plane, and in the \(y\)-\)z\) plane.

**Ex 12.1.6** Find the lengths of the sides of the triangle with vertices \((1,0,1)\), \((2,2,-1)\), and \((-3,2,-2)\). (answer)

**Ex 12.1.7** Find the lengths of the sides of the triangle with vertices \((2,2,3)\), \((8,6,5)\), and \((-1,0,2)\). Why do the results tell you that this isn't really a triangle? (answer)

**Ex 12.1.8** Find an equation of the sphere with center at \((1,1,1)\) and radius 2. (answer)

**Ex 12.1.9** Find an equation of the sphere with center at \((2,-1,3)\) and radius 5. (answer)

**Ex 12.1.10** Find an equation of the sphere with center \((3, -2, 1)\) and that goes through the point \((4, 2, 5)\).

**Ex 12.1.11** Find an equation of the sphere with center at \((2,1,-1)\) and radius 4. Find an equation for the intersection of this sphere with the \(y\)-\)z\) plane; describe this intersection geometrically. (answer)

**Ex 12.1.12** Consider the sphere of radius 5 centered at \((2,3,4)\). What is the intersection of this sphere with each of the coordinate planes?

**Ex 12.1.13** Show that for all values of \(\theta\) and \(\phi\), the point \((a\sin\phi\cos\theta,a\sin\phi\sin\theta,a\cos\phi)\) lies on the sphere given by \( x^2+y^2+z^2=a^2\).

**Ex 12.1.14** Prove that the midpoint of the line segment connecting \( (x_1,y_1,z_1)\) to \( (x_2,y_2,z_2)\) is at \(\ds\left({x_1+x_2\over 2},{y_1+y_2\over 2},{z_1+z_2\over 2}\right)\).

**Ex 12.1.15** Any three points \(P_1(x_1,y_1,z_1)\), \(P_2(x_2,y_2,z_2)\), \(P_3(x_3,y_3,z_3)\), lie in a plane and form a triangle. The **triangle inequality** says that \( d(P_1,P_3)\le d(P_1,P_2)+d(P_2,P_3)\). Prove the triangle inequality using either algebra (messy) or the law of cosines (less messy).

**Ex 12.1.16** Is it possible for a plane to intersect a sphere in exactly two points? Exactly one point? Explain.

## 12.2: Vectors

**Ex 12.2.1** Draw the vector \(\langle 3,-1\rangle\) with its tail at the origin.

**Ex 12.2.2** Draw the vector \(\langle 3,-1,2\rangle\) with its tail at the origin.

**Ex 12.2.3** Let \({\bf A}\) be the vector with tail at the origin and head at \((1,2)\); let \({\bf B}\) be the vector with tail at the origin and head at \((3,1)\). Draw \({\bf A}\) and \({\bf B}\) and a vector \({\bf C}\) with tail at \((1,2)\) and head at \((3,1)\). Draw \(\bf C\) with its tail at the origin.

**Ex 12.2.4** Let \({\bf A}\) be the vector with tail at the origin and head at \((-1,2)\); let \({\bf B}\) be the vector with tail at the origin and head at \((3,3)\). Draw \({\bf A}\) and \({\bf B}\) and a vector \({\bf C}\) with tail at \((-1,2)\) and head at \((3,3)\). Draw \(\bf C\) with its tail at the origin.

**Ex 12.2.5** Let \({\bf A}\) be the vector with tail at the origin and head at \((5,2)\); let \({\bf B}\) be the vector with tail at the origin and head at \((1,5)\). Draw \({\bf A}\) and \({\bf B}\) and a vector \({\bf C}\) with tail at \((5,2)\) and head at \((1,5)\). Draw \(\bf C\) with its tail at the origin.

**Ex 12.2.6** Find \(|{\bf v}|\), \({\bf v}+{\bf w}\), \({\bf v}-{\bf w}\), \(|{\bf v}+{\bf w}|\), \(|{\bf v}-{\bf w}|\) and \(-2{\bf v}\) for \({\bf v} = \langle 1,3\rangle\) and \({\bf w} = \langle -1,-5\rangle\). (answer)

**Ex 12.2.7** Find \(|{\bf v}|\), \({\bf v}+{\bf w}\), \({\bf v}-{\bf w}\), \(|{\bf v}+{\bf w}|\), \(|{\bf v}-{\bf w}|\) and \(-2{\bf v}\) for \({\bf v} = \langle 1,2,3\rangle\) and \({\bf w} = \langle -1,2,-3\rangle\). (answer)

**Ex 12.2.8** Find \(|{\bf v}|\), \({\bf v}+{\bf w}\), \({\bf v}-{\bf w}\), \(|{\bf v}+{\bf w}|\), \(|{\bf v}-{\bf w}|\) and \(-2{\bf v}\) for \({\bf v} = \langle 1,0,1\rangle\) and \({\bf w} = \langle -1,-2,2 \rangle\). (answer)

**Ex 12.2.9** Find \(|{\bf v}|\), \({\bf v}+{\bf w}\), \({\bf v}-{\bf w}\), \(|{\bf v}+{\bf w}|\), \(|{\bf v}-{\bf w}|\) and \(-2{\bf v}\) for \({\bf v} = \langle 1,-1,1\rangle\) and \({\bf w} = \langle 0,0,3\rangle\). (answer)

**Ex 12.2.10** Find \(|{\bf v}|\), \({\bf v}+{\bf w}\), \({\bf v}-{\bf w}\), \(|{\bf v}+{\bf w}|\), \(|{\bf v}-{\bf w}|\) and \(-2{\bf v}\) for \({\bf v} = \langle 3,2,1\rangle\) and \({\bf w} = \langle -1,-1,-1\rangle\). (answer)

**Ex 12.2.11** Let \(P=(4,5,6)\), \(Q=(1,2,-5)\). Find \( \overrightarrow{\strut PQ}\). Find a vector with the same direction as \( \overrightarrow{\strut PQ}\) but with length 1. Find a vector with the same direction as \( \overrightarrow{\strut PQ}\) but with length 4. (answer)

**Ex 12.2.12** If \(A, B\), and \(C\) are three points, find \( \overrightarrow{\strut AB}+ \overrightarrow{\strut BC}+ \overrightarrow{\strut CA}\). (answer)

**Ex 12.2.13** Consider the 12 vectors that have their tails at the center of a clock and their respective heads at each of the 12 digits. What is the sum of these vectors? What if we remove the vector corresponding to 4 o'clock? What if, instead, all vectors have their tails at 12 o'clock, and their heads on the remaining digits? (answer)

**Ex 12.2.14** Let \(\bf a\) and \(\bf b\) be nonzero vectors in two dimensions that are not parallel or anti-parallel. Show, algebraically, that if \(\bf c\) is any two dimensional vector, there are scalars \(s\) and \(t\) such that \({\bf c}=s{\bf a}+t{\bf b}\).

**Ex 12.2.15** Does the statement in the previous exercise hold if the vectors \(\bf a\), \(\bf b\), and \(\bf c\) are three dimensional vectors? Explain.

## 12.3: The Dot Product

**Ex 12.3.1** Find \(\langle 1,1,1\rangle\cdot\langle 2,-3,4\rangle\). (answer)

**Ex 12.3.2** Find \(\langle 1,2,0\rangle\cdot\langle 0,0,57\rangle\). (answer)

**Ex 12.3.3** Find \(\langle 3,2,1\rangle\cdot\langle 0,1,0\rangle\). (answer)

**Ex 12.3.4** Find \(\langle -1,-2,5\rangle\cdot\langle 1,0,-1 \rangle\). (answer)

**Ex 12.3.5** Find \(\langle 3,4,6\rangle\cdot\langle 2,3,4\rangle\). (answer)

**Ex 12.3.6** Find the cosine of the angle between \(\langle 1,2,3\rangle\) and \(\langle 1,1,1\rangle\); use a calculator if necessary to find the angle. (answer)

**Ex 12.3.7** Find the cosine of the angle between \(\langle -1, -2,-3\rangle\) and \(\langle 5,0,2\rangle\); use a calculator if necessary to find the angle. (answer)

**Ex 12.3.8** Find the cosine of the angle between \(\langle 47,100,0\rangle\) and \(\langle 0,0,5\rangle\); use a calculator if necessary to find the angle. (answer)

**Ex 12.3.9** Find the cosine of the angle between \(\langle 1,0,1 \rangle\) and \(\langle 0,1,1\rangle\); use a calculator if necessary to find the angle. (answer)

**Ex 12.3.10** Find the cosine of the angle between \(\langle 2,0,0\rangle\) and \(\langle -1,1,-1\rangle\); use a calculator if necessary to find the angle. (answer)

**Ex 12.3.11** Find the angle between the diagonal of a cube and one of the edges adjacent to the diagonal. (answer)

**Ex 12.3.12** Find the scalar and vector projections of \(\langle 1,2,3\rangle\) onto \(\langle 1,2,0\rangle\). (answer)

**Ex 12.3.13** Find the scalar and vector projections of \(\langle 1,1,1\rangle\) onto \(\langle 3,2,1\rangle\). (answer)

**Ex 12.3.14** A force of 10 pounds is applied to a wagon, directed at an angle of \( 30^\circ\). Find the component of this force pulling the wagon straight up, and the component pulling it horizontally along the ground. (answer)

**Ex 12.3.15** A force of 15 pounds is applied to a wagon, directed at an angle of \( 45^\circ\). Find the component of this force pulling the wagon straight up, and the component pulling it horizontally along the ground. (answer)

**Ex 12.3.16** Use the dot product to find a non-zero vector \({\bf w}\) perpendicular to both \({\bf u}=\langle 1,2,-3\rangle\) and \({\bf v}=\langle 2,0,1\rangle\). (answer)

**Ex 12.3.17** Let \({\bf x}=\langle 1,1,0 \rangle\) and \({\bf y}=\langle 2,4,2 \rangle\). Find a unit vector that is perpendicular to both \(\bf x\) and \(\bf y\). (answer)

**Ex 12.3.18** Do the three points \((1,2,0)\), \((-2,1,1)\), and \((0,3,-1)\) form a right triangle? (answer)

**Ex 12.3.19** Do the three points \((1,1,1)\), \((2,3,2)\), and \((5,0,-1)\) form a right triangle? (answer)

**Ex 12.3.20** Show that \(|{\bf A}\cdot{\bf B}|\le|{\bf A}||{\bf B}|\)

**Ex 12.3.21** Let \(\bf x\) and \(\bf y\) be perpendicular vectors. Use Theorem 12.3.5 to prove that \( |{\bf x}|^2+|{\bf y}|^2=|{\bf x}+{\bf y}|^2\). What is this result better known as?

**Ex 12.3.22** Prove that the diagonals of a rhombus intersect at right angles.

**Ex 12.3.23** Suppose that \({\bf z}=|{\bf x}| {\bf y} + |{\bf y}| {\bf x}\) where \(\bf x\), \(\bf y\), and \(\bf z\) are all nonzero vectors. Prove that \(\bf z\) bisects the angle between \(\bf x\) and \(\bf y\).

**Ex 12.3.24** Prove Theorem 12.3.5.

## 12.4: The Cross Product

**Ex 12.4.1** Find the cross product of \(\langle 1,1,1\rangle\) and \(\langle 1,2,3\rangle\). (answer)

**Ex 12.4.2** Find the cross product of \(\langle 1,0,2\rangle\) and \(\langle -1,-2,4\rangle\). (answer)

**Ex 12.4.3** Find the cross product of \(\langle -2,1,3\rangle\) and \(\langle 5,2,-1\rangle\). (answer)

**Ex 12.4.4** Find the cross product of \(\langle 1,0,0\rangle\) and \(\langle 0,0,1\rangle\). (answer)

**Ex 12.4.5** Two vectors \({\bf u}\) and \({\bf v}\) are separated by an angle of \(\pi/6\), and \(|{\bf u}|=2\) and \(|{\bf v}|=3\). Find \(|{\bf u}\times{\bf v}|\). (answer)

**Ex 12.4.6** Two vectors \({\bf u}\) and \({\bf v}\) are separated by an angle of \(\pi/4\), and \(|{\bf u}|=3\) and \(|{\bf v}|=7\). Find \(|{\bf u}\times{\bf v}|\). (answer)\

**Ex 12.4.7** Find the area of the parallelogram with vertices \((0,0)\), \((1,2)\), \((3,7)\), and \((2,5)\). (answer)

**Ex 12.4.8** Find and explain the value of \(({\bf i} \times {\bf j}) \times {\bf k}\) and \(({\bf i} + {\bf j}) \times ({\bf i} - {\bf j})\).

**Ex 12.4.9** Prove that for all vectors \({\bf u}\) and \({\bf v}\), \(({\bf u}\times{\bf v})\cdot{\bf v}=0\).

**Ex 12.4.10** Prove Theorem 12.4.1.

**Ex 12.4.11** Define the triple product of three vectors, \({\bf x}\), \({\bf y}\), and \({\bf z}\), to be the scalar \({\bf x} \cdot ({\bf y} \times {\bf z})\). Show that three vectors lie in the same plane if and only if their triple product is zero. Verify that \(\langle 1, 5, -2 \rangle\), \(\langle 4, 3, 0 \rangle\) and \(\langle 6, 13, -4 \rangle\) are coplanar.

## 12.5: Lines and Planes

**Ex 12.5.1** Find an equation of the plane containing \((6,2,1)\) and perpendicular to \(\langle 1,1,1\rangle\). (answer)

**Ex 12.5.2** Find an equation of the plane containing \((-1,2,-3)\) and perpendicular to \(\langle 4,5,-1\rangle\). (answer)

**Ex 12.5.3** Find an equation of the plane containing \((1,2,-3)\), \((0,1,-2)\) and \((1,2,-2)\). (answer)

**Ex 12.5.4** Find an equation of the plane containing \((1,0,0)\), \((4,2,0)\) and \((3,2,1)\). (answer)

**Ex 12.5.5** Find an equation of the plane containing \((1,0,0)\) and the line \(\langle 1,0,2\rangle + t\langle 3,2,1\rangle\). (answer)

**Ex 12.5.6** Find an equation of the plane containing the line of intersection of \(x+y+z=1\) and \(x-y+2z=2\), and perpendicular to the \(x\)-\)y\) plane. (answer)

**Ex 12.5.7** Find an equation of the line through \((1,0,3)\) and \((1,2,4)\). (answer)

**Ex 12.5.8** Find an equation of the line through \((1,0,3)\) and perpendicular to the plane \(x+2y-z=1\). (answer)

**Ex 12.5.9** Find an equation of the line through the origin and perpendicular to the plane \(x+y-z=2\). (answer)

**Ex 12.5.10** Find \(a\) and \(c\) so that \((a,1,c)\) is on the line through \((0,2,3)\) and \((2,7,5)\). (answer)

**Ex 12.5.11** Explain how to discover the solution in example 12.5.5.

**Ex 12.5.12** Determine whether the lines \(\langle 1,3,-1\rangle+t\langle 1,1,0\rangle\) and \(\langle 0,0,0\rangle+t\langle 1,4,5\rangle\) are parallel, intersect, or neither. (answer)

**Ex 12.5.13** Determine whether the lines \(\langle 1,0,2\rangle+t\langle -1,-1,2\rangle\) and \(\langle 4,4,2\rangle+t\langle 2,2,-4\rangle\) are parallel, intersect, or neither. (answer)

**Ex 12.5.14** Determine whether the lines \(\langle 1,2,-1\rangle+t\langle 1,2,3\rangle\) and \(\langle 1,0,1\rangle+t\langle 2/3,2,4/3\rangle\) are parallel, intersect, or neither. (answer)

**Ex 12.5.15** Determine whether the lines \(\langle 1,1,2\rangle+t\langle 1,2,-3\rangle\) and \(\langle 2,3,-1\rangle+t\langle 2,4,-6\rangle\) are parallel, intersect, or neither. (answer)

**Ex 12.5.16** Find a unit normal vector to each of the coordinate planes.

**Ex 12.5.17** Show that \(\langle 2,1,3 \rangle + t \langle 1,1,2 \rangle\) and \(\langle 3, 2, 5 \rangle + s \langle 2, 2, 4 \rangle\) are the same line.

**Ex 12.5.18** Give a prose description for each of the following processes:

- Given two distinct points, find the line that goes through them.
- Given three points (not all on the same line), find the plane that goes through them. Why do we need the caveat that not all points be on the same line?
- Given a line and a point not on the line, find the plane that contains them both.
- Given a plane and a point not on the plane, find the line that is perpendicular to the plane through the given point.

**Ex 12.5.19** Find the distance from \((2,2,2)\) to \(x+y+z=-1\). (answer)

**Ex 12.5.20** Find the distance from \((2,-1,-1)\) to \(2x-3y+z=2\). (answer)

**Ex 12.5.21** Find the distance from \((2,-1,1)\) to \(\langle 2,2,0\rangle+t\langle 1,2,3\rangle\). (answer)

**Ex 12.5.22** Find the distance from \((1,0,1)\) to \(\langle 3,2,1\rangle+t\langle 2,-1,-2\rangle\). (answer)

**Ex 12.5.23** Find the cosine of the angle between the planes \(x+y+z=2\) and \(x+2y+3z=8\). (answer)

**Ex 12.5.24** Find the cosine of the angle between the planes \(x-y+2z=2\) and \(3x-2y+z=5\). (answer)

## 12.6: Other Coordinate Systems

**Ex 12.6.1** Convert the following points in rectangular coordinates to cylindrical and spherical coordinates:

- a. \((1,1,1)\) b. \((7,-7,5)\) c. \((\cos(1),\sin(1),1)\) d. \((0,0,-\pi)\) (answer)

**Ex 12.6.2** Find an equation for the sphere \( x^2+y^2+z^2=4\) in cylindrical coordinates. (answer)

**Ex 12.6.3** Find an equation for the \(y\)-\)z\) plane in cylindrical coordinates. (answer)

**Ex 12.6.4** Find an equation equivalent to \( x^2+y^2+2z^2+2z-5=0\) in cylindrical coordinates. (answer)

**Ex 12.6.5** Suppose the curve \( z=e^{-x^2}\) in the \(x\)-\)z\) plane is rotated around the \(z\) axis. Find an equation for the resulting surface in cylindrical coordinates. (answer)

**Ex 12.6.6** Suppose the curve \( z=x\) in the \(x\)-\)z\) plane is rotated around the \(z\) axis. Find an equation for the resulting surface in cylindrical coordinates. (answer)

**Ex 12.6.7** Find an equation for the plane \(y=0\) in spherical coordinates. (answer)

**Ex 12.6.8** Find an equation for the plane \(z=1\) in spherical coordinates. (answer)

**Ex 12.6.9** Find an equation for the sphere with radius 1 and center at \((0,1,0)\) in spherical coordinates. (answer)

**Ex 12.6.10** Find an equation for the cylinder \( x^2+y^2=4\) in spherical coordinates. (answer)

**Ex 12.6.11** Suppose the curve \( z=x\) in the \(x\)-\)z\) plane is rotated around the \(z\) axis. Find an equation for the resulting surface in spherical coordinates. (answer)

**Ex 12.6.12** Plot the polar equations \(r=\sin(\theta)\) and \(r=\cos(\theta)\) and comment on their similarities. (If you get stuck on how to plot these, you can multiply both sides of each equation by \(r\) and convert back to rectangular coordinates).

**Ex 12.6.13** Extend exercises 6 and 11 by rotating the curve \(z=mx\) around the \(z\) axis and converting to both cylindrical and spherical coordinates. (answer)

**Ex 12.6.14** Convert the spherical formula \(\rho=\sin \theta \sin \phi\) to rectangular coordinates and describe the surface defined by the formula (Hint: Multiply both sides by \(\rho\).) (answer)

**Ex 12.6.15** We can describe points in the first octant by \(x >0\), \(y>0\) and \(z>0\). Give similar inequalities for the first octant in cylindrical and spherical coordinates. (0\); \(0 < \theta < \pi/2\), \(r>0\), \(z>0\)">answer)