Exercises for Section 11.1

18

(Exercises with colored numbers are solved in the Study Guide.)

Complete the computations in Exercises 1 to 4.

Question 11.6

\((-21,23)-(?,6)=(-25,?)\)

Question 11.7

\(3(133,-0.33,0)+(-399,0.99,0)=(?,?,?)\)

Question 11.8

\((8a,-2b,13c)=(52,12,11)+\frac{1}{2}(?,?,?)\)

Question 11.9

\((2,3,5)-4{\bf i}+3{\bf j}=(?,?,?)\)

In Exercises 5 to 8, sketch the given vectors \({\bf v}\) and \({\bf w}\). On your sketch, draw in \(-{\bf v},{\bf v}+{\bf w},\) and \({\bf v}-{\bf w}\).

Question 11.10

\({\bf v}=(2,1)\) and \({\bf w}=(1,2)\)

Question 11.11

\({\bf v}=(0,4)\) and \({\bf w}=(2,-1)\)

Question 11.12

\({\bf v}=(2,3,-6)\) and \({\bf w}=(-1,1,1)\)

Question 11.13

\({\bf v}=(2,1,3)\) and \({\bf w}=(-2,0,-1)\)

Question 11.14

Let \(\textbf{v}=2\textbf{i}+\textbf{j}\) and \(\textbf{w}=\textbf{i}+2\textbf{j}\) Sketch \(\textbf{v}, \ \textbf{w}, \ \textbf{v}+\textbf{w}, \ 2\textbf{w},\) and \(\textbf{v}-\textbf{w}\) in the plane.

Question 11.15

Sketch (1, \(-2\), 3) and \((-\frac{1}{3}, \frac{2}{3}, -1)\) Why do these vectors point in opposite directions?

Question 11.16

What restrictions must be made on \(x,y\), and \(z\) so that the triple \((x,y,z)\) will represent a point on the \(y\) axis? On the \(z\) axis? In the \(xz\) plane? In the \(yz\) plane?

Question 11.17

  • (a) Generalize the geometric construction in Figure 11.9 to show that if \({\bf v}_1=(x,y,z)\) and \({\bf v}_2=(x',y',z')\), then \({\bf v}_1+ {\bf v}_2 = (x+x',y+y',z+z')\).
  • (b) Using an argument based on similar triangles, prove that \(\alpha {\bf v}= (\alpha x,\alpha y,\alpha z)\) when \({\bf v}=\) \((x,y,z)\).

In Exercises 13 to 19, use set theoretic or vector notation or both to describe the points that lie in the given configurations.

Question 11.18

The plane spanned by \({\bf v}_1=(2,7,0)\) and \({\bf v}_2=(0,2,7)\)

Question 11.19

The plane spanned by \({\bf v}_1=(3,-1,1)\) and \({\bf v}_2=(0,3,4)\)

Question 11.20

The line passing through \((-1,-1,-1)\) in the direction of \({\bf j}\)

Question 11.21

The line passing through \((0, 2, 1)\) in the direction of \(2{\bf i}-{\bf k}\)

Question 11.22

The line passing through \((-1,-1,-1)\) and \((1,-1,2)\)

Question 11.23

The line passing through \((-5,0,4)\) and \((6,-3,2)\)

Question 11.24

The parallelogram whose adjacent sides are the vectors \({\bf i}+3{\bf k}\) and \(-2{\bf j}\)

Question 11.25

Show that \( {\bf l}_1(t)=(1,2,3) +t(1, 0, -2)\) and \( {\bf l}_2(t)=(2, 2, 1) +t(-2, 0, 4)\) parametrize the same line.

Question 11.26

Do the points \((2, 3, -4), (2, 1, -1)\), and \((2, 7, -10)\) lie on the same line?

Question 11.27

Let \(\textbf{u}=(1, 2), \textbf{v}=(-3, 4)\), and \(\textbf{w}=(5, 0)\):

  • (a) Draw these vectors in \(\mathbb{R}^2\)
  • (b) Find scalars \(\lambda_1\) and \(\lambda_2\) such that \(\textbf{w}=\lambda_1\textbf{u} +\lambda_2\textbf{v}\)

Question 11.28

Suppose \(A, B,\) and \(C\) are vertices of a triangle. Find \(\overrightarrow{AB} +\overrightarrow{BC} +\overrightarrow{CA}\).

Question 11.29

Find the points of intersection of the line \(x=3+2t,y=7+8t,z=-2+t\), that is, \({\bf l}(t)=\) \((3+2t,7+8t,-2+t)\), with the coordinate planes.

Question 11.30

Show that there are no points \((x,y,z)\) satisfying \(2x-3y+z-2=0\) and lying on the line \({\bf v}=(2,-2,-1)+t(1,1,1)\).

Question 11.31

Show that every point on the line \({\bf v}=(1,-1,2) + t(2,3,1)\) satisfies the equation \(5x-3y-z-6=0.\)

Question 11.32

Determine whether the lines \(x = 3t + 2, y = t - 1, z = 6t + 1\), and \(x = 3s-1, y = s-2, z = s\) intersect.

Question 11.33

Do the lines \((x,y,z)=(t + 4, 4t + 5, t-2)\) and \((x,y,z)=(2s+3, s+1, 2s-3)\) intersect?

19

In Exercises 29 to 31, use vector methods to describe the given configurations.

Question 11.34

The parallelepiped with edges the vectors \({\bf a}, {\bf b}\), and \({\bf c}\) emanating from the origin

Question 11.35

The points within the parallelogram with one corner at \((x_0,y_0,z_0)\) whose sides extending from that corner are equal in magnitude and direction to vectors \({\bf a}\) and \({\bf b}\)

Question 11.36

The plane determined by the three points \((x_0,y_0,z_0),(x_1,y_1,z_1)\), and \((x_2,y_2,z_2)\)

Prove the statements in Exercises 32 to 34.

Question 11.37

The line segment joining the midpoints of two sides of a triangle is parallel to and has half the length of the third side.

Question 11.38

If PQR is a triangle in space and \(b>0\) is a number, then there is a triangle with sides parallel to those of PQR and side lengths \(b\) times those of PQR.

Question 11.39

The medians of a triangle intersect at a point, and this point divides each median in a ratio of \(2\,{:}\,1\).

Problems 35 and 36 require some knowledge of chemical notation.

Question 11.40

Write the chemical equation \({\rm CO} + {\rm H}_2{\rm O} = {\rm H}_2 + {\rm CO}_2\) as an equation in ordered triples \((x_1,x_2,x_3)\), where \(x_1,x_2,x_3\) are the number of carbon, hydrogen, and oxygen atoms, respectively, in each molecule.

Question 11.41

  • (a) Write the chemical equation \(p{\rm C}_3{\rm H}_4{\rm O}_3+q{\rm O}_2 = r{\rm CO}_2+s{\rm H}_2{\rm O}\) as an equation in ordered triples with unknown coefficients \(p,q,r\), and \(s\)
  • (b) Find the smallest positive integer solution for \(p,q,r\), and \(s\)
  • (c) Illustrate the solution by a vector diagram in space.

Question 11.42

Find a line that lies entirely in the set defined by the equation \(x^2+y^2-z^2=1\)