Chapter 12

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.

Find \(|\boldsymbol{v}|, \boldsymbol{v}+\boldsymbol{w}, \boldsymbol{v}-\boldsymbol{w},|\boldsymbol{v}+\boldsymbol{w}|,|\boldsymbol{v}-\boldsymbol{w}|\) and \(-2 \boldsymbol{v}\) for \(\boldsymbol{v}=\langle 1,-1,1\rangle\) and \(\boldsymbol{w}=\langle 0,0,3\rangle .\)

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

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

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.

Find \(|\boldsymbol{v}|, \boldsymbol{v}+\boldsymbol{w}, \boldsymbol{v}-\boldsymbol{w},|\boldsymbol{v}+\boldsymbol{w}|,|\boldsymbol{v}-\boldsymbol{w}|\) and \(-2 \boldsymbol{v}\) for \(\boldsymbol{v}=\langle 3,2,1\rangle\) and \(\boldsymbol{w}=\langle-1,-1,-1\rangle .\)

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

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).

Define the triple product of three vectors, \(\boldsymbol{x}, \boldsymbol{y},\) and \(z,\) to be the scalar \(\boldsymbol{x} \cdot(\boldsymbol{y} \times \boldsymbol{z}) .\) 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 all lie in the same plane.

Find the angle between the diagonal of a cube and one of the edges adjacent to the diagonal.

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

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 yz-plane; describe this intersection geometrically.

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.

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

If \(A, B,\) and \(C\) are three points, find \(\overrightarrow{A B}+\overrightarrow{B C}+\overrightarrow{C A} .\)

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

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 \(\rho .)\)

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 paral lel, intersect, or neither.

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

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?

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 \(x^{2}+y^{2}+z^{2}=a^{2}\).

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.

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

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.

Let a and b be nonzero vectors in two dimensions that are not parallel or anti-parallel. (Vectors are parallel if they point in the same direction, anti-parallel if they point in opposite directions.) Show, algebraically, that if \(\boldsymbol{c}\) is any two dimensional vector, there are scalars \(\mathrm{s}\) and \(t\) such that \(\boldsymbol{c}=\mathrm{sa}+\mathrm{tb}\).

Prove that the midpoint of the line segment connecting \(\left(x_{1}, y_{1}, z_{1}\right)\) to \(\left(x_{2}, y_{2}, z_{2}\right)\) is at \(\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}, \frac{z_{1}+z_{2}}{2}\right)\)

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.

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.

Any three points \(P_{1}\left(x_{1}, y_{1}, z_{1}\right), P_{2}\left(x_{2}, y_{2}, z_{2}\right), P_{3}\left(x_{3}, y_{3}, z_{3}\right),\) lie in a plane and form a triangle. The triangle inequality says that \(d\left(P_{1}, P_{3}\right) \leq d\left(P_{1}, P_{2}\right)+d\left(P_{2}, P_{3}\right) .\) Prove the triangle inequality using either algebra (messy) or the law of cosines (less messy).

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

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

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

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.

Let \(x=\langle 1,1,0\rangle\) and \(y=\langle 2,4,2\rangle .\) Find a unit vector that is perpendicular to both \(x\) and \(y\)

Exercise 12.5.18 Give a prose description for each of the following processes: (a) Given two distinct points, find the line that goes through them. (b) 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? (c) Given a line and a point not on the line, find the plane that contains them both. (d) Given a plane and a point not on the plane, find the line that is perpendicular to the plane through the given point.

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

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

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

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

Show that \(|\boldsymbol{v} \cdot \boldsymbol{w}| \leq|\boldsymbol{v}||\boldsymbol{w}|\).

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

Let \(x\) and \(y\) be perpendicular vectors. Use Theorem 12.6 to prove that \(|x|^{2}+|y|^{2}=\) \(|\boldsymbol{x}+\boldsymbol{y}|^{2}\). What is this result better known as?

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

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

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

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

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