18
(Exercises with colored numbers are solved in the Study Guide.)
Complete the computations in Exercises 1 to 4.
\((-21,23)-(?,6)=(-25,?)\)
\(3(133,-0.33,0)+(-399,0.99,0)=(?,?,?)\)
\((8a,-2b,13c)=(52,12,11)+\frac{1}{2}(?,?,?)\)
\((2,3,5)-4{\bf i}+3{\bf j}=(?,?,?)\)
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 \({\bf v}-{\bf w}\).
\({\bf v}=(2,1)\) and \({\bf w}=(1,2)\)
\({\bf v}=(0,4)\) and \({\bf w}=(2,-1)\)
\({\bf v}=(2,3,-6)\) and \({\bf w}=(-1,1,1)\)
\({\bf v}=(2,1,3)\) and \({\bf w}=(-2,0,-1)\)
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.
Sketch (1, \(-2\), 3) and \((-\frac{1}{3}, \frac{2}{3}, -1)\) Why do these vectors point in opposite directions?
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?
In Exercises 13 to 19, use set theoretic or vector notation or both to describe the points that lie in the given configurations.
The plane spanned by \({\bf v}_1=(2,7,0)\) and \({\bf v}_2=(0,2,7)\)
The plane spanned by \({\bf v}_1=(3,-1,1)\) and \({\bf v}_2=(0,3,4)\)
The line passing through \((-1,-1,-1)\) in the direction of \({\bf j}\)
The line passing through \((0, 2, 1)\) in the direction of \(2{\bf i}-{\bf k}\)
The line passing through \((-1,-1,-1)\) and \((1,-1,2)\)
The line passing through \((-5,0,4)\) and \((6,-3,2)\)
The parallelogram whose adjacent sides are the vectors \({\bf i}+3{\bf k}\) and \(-2{\bf j}\)
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.
Do the points \((2, 3, -4), (2, 1, -1)\), and \((2, 7, -10)\) lie on the same line?
Let \(\textbf{u}=(1, 2), \textbf{v}=(-3, 4)\), and \(\textbf{w}=(5, 0)\):
Suppose \(A, B,\) and \(C\) are vertices of a triangle. Find \(\overrightarrow{AB} +\overrightarrow{BC} +\overrightarrow{CA}\).
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.
Show that there are no points \((x,y,z)\) satisfying \(2x-3y+z-2=0\) and lying on the line \({\bf v}=(2,-2,-1)+t(1,1,1)\).
Show that every point on the line \({\bf v}=(1,-1,2) + t(2,3,1)\) satisfies the equation \(5x-3y-z-6=0.\)
Determine whether the lines \(x = 3t + 2, y = t - 1, z = 6t + 1\), and \(x = 3s-1, y = s-2, z = s\) intersect.
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.
The parallelepiped with edges the vectors \({\bf a}, {\bf b}\), and \({\bf c}\) emanating from the origin
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 \({\bf b}\)
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.
The line segment joining the midpoints of two sides of a triangle is parallel to and has half the length of the third side.
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.
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.
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.
Find a line that lies entirely in the set defined by the equation \(x^2+y^2-z^2=1\)