Chapter 7

\(\mathbf{a}=\langle 1,-3,2\rangle, \mathbf{b}=\langle-1,1,1\rangle\), and \(\mathbf{c}=\langle 2,6,9\rangle .\) Find the indicated vector or scalar. \(\|\mathbf{b}\| \mathbf{a}+\|\mathbf{a}\| \mathbf{b}\)

Find the area of the triangle determined by the given points. $$ P_{1}(0,0,0), P_{2}(0,1,2), P_{3}(2,2,0) $$

Using vectors, show that the diagonals of a parallelogram bisect each other. [Hint: Let \(M\) be the midpoint of one diagonal and \(N\) the midpoint of the other.]

In Problems, find, if possible, an equation of a plane that contains the given points. $$ (1,2,-1),(4,3,1),(7,4,3) $$

Find a unit vector in the opposite direction of \(\mathbf{a}=\langle 10,-5,10\rangle\).

Find the area of the triangle determined by the given points. $$ P_{1}(1,2,4), P_{2}(1,-1,3), P_{3}(-1,-1,2) $$

Using vectors, show that the line segment between the midpoints of two sides of a triangle is parallel to the third side and half as long.

Use the dot product to provethe Cauchy-Schwarz inequality: \(|\mathbf{a} \cdot \mathbf{b}| \leq\|\mathbf{a}\|\|\mathbf{b}\|\).

In Problems, find, if possible, an equation of a plane that contains the given points. $$ (2,1,2),(4,1,0),(5,0,-5) $$

Find a unit vector in the same direction as \(\mathbf{a}=\mathbf{i}-3 \mathbf{j}+2 \mathbf{k}\).

Find the area of the triangle determined by the given points. $$ P_{1}(1,0,3), P_{2}(0,0,6), P_{3}(2,4,5) $$

Use the dot product to prove the triangle inequality \(\|\mathbf{a}+\mathbf{b}\| \leq\|\mathbf{a}\|+\|\mathbf{b}\| .\) [Hint: Consider property \((v i)\) of the dot product.]

In Problems, find an equation of the plane that satisfies the given conditions. Contains \((2,3,-5)\) and is parallel to \(x+y-4 z=1\)

Find a vector \(\mathbf{b}\) that is four times as long as \(\mathbf{a}=\mathbf{i}-\mathbf{j}+\mathbf{k}\) in the same direction as a.

Find the volume of the parallelepiped for which the given vectors are three edges. $$ \mathbf{a}=\mathbf{i}+\mathbf{j}, \mathbf{b}=-\mathbf{i}+4 \mathbf{j}, \mathbf{c}=2 \mathbf{i}+2 \mathbf{j}+2 \mathbf{k} $$

Prove that the vectoe \(\mathbf{n}=a \mathbf{i}+b \mathbf{j}\) is perpendicular to the line whose equation is \(a x+b y+c=0 .\) [Hint: Let \(P_{1}\left(x_{1}, y_{1}\right)\) and \(P_{2}\left(x_{2}, y_{2}\right)\) be distinct points on the line.]Prove that the vectoe \(\mathbf{n}=a \mathbf{i}+b \mathbf{j}\) is perpendicular to the line whose equation is \(a x+b y+c=0 .\) [Hint: Let \(P_{1}\left(x_{1}, y_{1}\right)\) and \(P_{2}\left(x_{2}, y_{2}\right)\) be distinct points on the line.]

In Problems, find an equation of the plane that satisfies the given conditions. Contains the origin and is parallel to \(5 x-y+z=6\)

Find a vector \(\mathbf{b}\) for which \(\|\mathbf{b}\|=\frac{1}{2}\) that is parallel to \(\mathbf{a}=\langle-6,3,-2\rangle\) but has the opposite direction.

Find the volume of the parallelepiped for which the given vectors are three edges. $$ \mathbf{a}=3 \mathbf{i}+\mathbf{j}+\mathbf{k}, \mathbf{b}=\mathbf{i}+4 \mathbf{j}+\mathbf{k}, \mathbf{c}=\mathbf{i}+\mathbf{j}+5 \mathbf{k} $$

In Problems, find an equation of the plane that satisfies the given conditions. Contains \((3,6,12)\) and is parallel to the \(x y\) -plane

Determine whether the vectors \(\mathbf{a}=4 \mathbf{i}+6 \mathbf{j}, \mathbf{b}=-2 \mathbf{i}+6 \mathbf{j}-6 \mathbf{k}\) and \(\mathbf{c}=\frac{5}{2} \mathbf{i}+3 \mathbf{j}+\frac{1}{2} \mathbf{k}\) are coplanar.

In Problems, find an equation of the plane that satisfies the given conditions. Contains \((-7,-5,18)\) and is perpendicular to the \(y\) -axis

Determine whether the four points \(P_{1}(1,1,-2), P_{2}(4,0,-3)\), \(P_{3}(1,-5,10)\), and \(P_{4}(-7,2,4)\) lie in the same plane.

In Problems, find an equation of the plane that satisfies the given conditions. Contains the lines \(x=1+3 t, y=1-t, z=2+t ;\) \(x=4+4 s, y=2 s, z=3+s\)

In Problems, find an equation of the plane that satisfies the given conditions. Contains the lines \(\frac{x-1}{2}=\frac{y+1}{-1}=\frac{z-5}{6}\) \(\mathbf{r}=\langle 1,-1,5\rangle+t\langle 1,1,-3\rangle\)

Two vectors a and b lie in the \(x z\) -plane so that the angle between them is \(120^{\circ} .\) If \(\|\mathbf{a}\|=\sqrt{27}\) and \(\|\mathbf{b}\|=8\), find all possible values of \(\mathbf{a} \times \mathbf{b}\).

In Problems, find an equation of the plane that satisfies the given conditions. Contains the parallel lines \(x=1+t, y=1+2 t, z=3+t ;\) \(x=3+s, y=2 s, z=-2+s\)

A three-dimensional lattice is a collection of integer combinations of three noncoplanar basis vectors \(\mathbf{a}, \mathbf{b}\), and \(\mathbf{c} .\) In crystallography, a lattice can specify the locations of atoms in a crystal. X-ray diffraction studies of crystals use the "reciprocal lattice" that has basis $$ \mathbf{A}=\frac{\mathbf{b} \times \mathbf{c}}{\mathbf{a} \cdot(\mathbf{b} \times \mathbf{c})}, \quad \mathbf{B}=\frac{\mathbf{c} \times \mathbf{a}}{\mathbf{b} \cdot(\mathbf{c} \times \mathbf{a})^{\prime}}, \quad \mathbf{C}=\frac{\mathbf{a} \times \mathbf{b}}{\mathbf{c} \cdot(\mathbf{a} \times \mathbf{b})} $$ (a) A certain lattice has basis vectors \(\mathbf{a}=\mathbf{i}, \mathbf{b}=\mathbf{j}\), and \(\mathbf{c}=\frac{1}{2}(\mathbf{i}+\mathbf{j}+\mathbf{k}) .\) Find basis vectors for the reciprocal lattice. (b) The unit cell of the reciprocal lattice is the parallelepiped with edges \(\mathbf{A}, \mathbf{B}\), and \(\mathbf{C}\), while the unit cell of the original lattice is the parallelepiped with edges \(\mathbf{a}, \mathbf{b}\), and \(\mathbf{c}\). Show that the volume of the unit cell of the reciprocal lattice is the reciprocal of the volume of the unit cell of the original lattice. [Hint: Start with \(\mathbf{B} \times \mathbf{C}\) and use (15).]

In Problems, find an equation of the plane that satisfies the given conditions. Contains the point \((4,0,-6)\) and the line \(x=3 t, y=2 t\), \(z=-2 t\)

In Problems, find an equation of the plane that satisfies the given conditions. Contains \((2,4,8)\) and is perpendicular to the line \(x=10-3 t\), \(y=5+t, z=6-\frac{1}{2} t\)

Prove \(a \times(\mathbf{b} \times \mathbf{c})=(\mathbf{a} \cdot \mathbf{c}) \mathbf{b}-(\mathbf{a} \cdot \mathbf{b}) \mathbf{c}\)

In Problems, find an equation of the plane that satisfies the given conditions. Contains \((1,1,1)\) and is perpendicular to the line through \((2,6,-3)\) and \((1,0,-2)\)

Prove or disprove \(\mathbf{a} \times(\mathbf{b} \times \mathbf{c})=(\mathbf{a} \times \mathbf{b}) \times \mathbf{c}\).

Let \(\mathscr{P}_{1}\) and \(\mathscr{P}_{2}\) be planes with normal vectors \(\mathbf{n}_{1}\) and \(\mathbf{n}_{2}\), respectively. \(\mathscr{P}_{1}\) and \(\Phi_{2}\) are orthogonal if \(\mathbf{n}_{1}\) and \(\mathbf{n}_{2}\) are orthogonal and parallel if \(\mathbf{n}_{1}\) and \(\mathbf{n}_{2}\) are parallel. Determine which of the following planes are orthogonal and which are parallel. (a) \(2 x-y+3 z=1\) (b) \(x+2 y+2 z=9\) (c) \(x+y-\frac{3}{2} z=2\) (d) \(-5 x+2 y+4 z=0\) (e) \(-8 x-8 y+12 z=1\) (f) \(-2 x+y-3 z=5\)

Prove \(\mathbf{a} \cdot(\mathbf{b} \times \mathbf{c})=(\mathbf{a} \times \mathbf{b}) \cdot \mathbf{c}\)

Find parametric equations for the line that contains \((-4,1,7)\) and is perpendicular to the plane \(-7 x+2 y+3 z=1\).

Prove \(\mathbf{a} \times(\mathbf{b} \times \mathbf{c})+\mathbf{b} \times(\mathbf{c} \times \mathbf{a})+\mathbf{c} \times(\mathbf{a} \times \mathbf{b})=\mathbf{0}\)

Determine which of the following planes are perpendicular to the line \(x=4-6 t, y=1+9 t, z=2+3 t\) (a) \(4 x+y+2 z=1\) (b) \(2 x-3 y+z=4\) (c) \(10 x-15 y-5 z=2\) (d) \(-4 x+6 y+2 z=9\)

Prove Lagrange's identity: $$ \|\mathbf{a} \times \mathbf{b}\|^{2}=\|\mathbf{a}\|^{2}\|\mathbf{b}\|^{2}-(\mathbf{a} \cdot \mathbf{b})^{2} . $$

Determine which of the following planes are parallel to the line \((1-x) / 2=(y+2) / 4=z-5\) (a) \(x-y+3 z=1\) (b) \(6 x-3 y=1\) (c) \(x-2 y+5 z=0\) (d) \(-2 x+y-2 z=7\)

Does \(\mathbf{a} \times \mathbf{b}=\mathbf{a} \times \mathbf{c}\) imply that \(\mathbf{b}=\mathbf{c}\) ?

In Problems, find parametric equations for the line of intersection of the given planes. $$ \begin{array}{r} 5 x-4 y-9 z=8 \\ x+4 y+3 z=4 \end{array} $$

Show that \((\mathbf{a}+\mathbf{b}) \times(\mathbf{a}-\mathbf{b})=2 \mathbf{b} \times \mathbf{a}\)

In Problems, find parametric equations for the line of intersection of the given planes. $$ \begin{array}{r} x+2 y-z=2 \\ 3 x-y+2 z=1 \end{array} $$

In Problems, find parametric equations for the line of intersection of the given planes. $$ \begin{array}{r} 4 x-2 y-z=1 \\ x+y+2 z=1 \end{array} $$

In Problems, find parametric equations for the line of intersection of the given planes. $$ \begin{aligned} 2 x-5 y+z &=0 \\ y &=0 \end{aligned} $$

In Problems, find the point of intersection of the given plane and line. $$ 2 x-3 y+2 z=-7 ; x=1+2 t, y=2-t, z=-3 t $$

In Problems, find the point of intersection of the given plane and line. $$ x+y+4 z=12 ; x=3-2 t, y=1+6 t, z=2-\frac{1}{2} t $$

In Problems, find the point of intersection of the given plane and line. $$ x+y-z=8 ; x=1, y=2, z=1+t $$

In Problems, find the point of intersection of the given plane and line. $$ x-3 y+2 z=0 ; x=4+t, y=2+t, z=1+5 t $$