Let \({\bf v} = 3 {\bf i} \,{+}\, 4 {\bf j} \,{+}\, 5 {\bf k}\) and \({\bf w} = {\bf i} - {\bf j} + {\bf k}\). Compute \({\bf v} + {\bf w}, 3 {\bf v}, 6 {\bf v} + 8 {\bf w}, -2 {\bf v}, {\bf v \,{\cdot}\, w}, {\bf v} \times {\bf w}\). Interpret each operation geometrically by graphing the vectors.
Repeat Exercise 1 with \({\bf v} = 2 {\bf j} + {\bf k}\) and \({\bf w} = - {\bf i} - {\bf k}\).
Find an equation for the plane containing the points (2, 1, \(-1\)), (3, 0, 2), and (4, \(-3\), 1).
Find an equation for a line that is parallel to the plane \(2x-3y+5z-10=0\) and passes through the point (\(-1\), 7, 4). (There are lots of them.)
Compute \({\bf v \,{\cdot}\, w}\) for the following sets of vectors:
Compute \({\bf v} \times {\bf w}\) for the vectors in Exercise 7. [Only part (b) is solved in the Study Guide.]
Find the cosine of the angle between the vectors in Exercise 7. [Only part (b) is solved in the Study Guide.]
Find the area of the parallelogram spanned by the vectors in Exercise 7. [Only part (b) is solved in the Study Guide.]
Use vector notation to describe the triangle in space whose vertices are the origin and the endpoints of vectors a and b.
Show that three vectors \({\bf a, b, c}\) lie in the same plane through the origin if and only if there are three scalars \(\alpha , \beta, \gamma\), not all zero, such that \(\alpha {\bf a} + \beta {\bf b} + \gamma {\bf c} ={\bf 0}\).
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For real numbers \(a_1, a_2 ,a_3, b_1, b_2, b_3\), show that \[ (a_1 b_1 + a_2 b_2 + a_3 b_3)^2 \le (a_1^2 +a_2^2 + a_3^2) (b_1^2 + b_2^2 + b_3^2). \]
Let \({\bf u,v,w}\) be unit vectors that are orthogonal to each other. If \({\bf a}= \alpha {\bf u} + \beta {\bf v} + \gamma {\bf w}\), show that \[ \alpha = {\bf a \,{\cdot}\, u}, \beta = {\bf a \,{\cdot}\, v}, \gamma= {\bf a \,{\cdot}\, w}. \] Interpret the results geometrically.
Find the products \(AB\) and \(BA\) where \[ A=\left[ \begin{array}{c@{\quad}c@{\quad}c} 1 & 5 & 2 \\ 0 & 2 & 3 \\ 1 & 0 & 2 \\ \end{array} \right] \quad \quad \quad B=\left[ \begin{array}{c@{\quad}c@{\quad}c} 2 & 0 & 1 \\ 1 & 3 & 0 \\ 2 & 4 & 1 \\ \end{array} \right]\!. \]
Find the products \(AB\) and \(BA\) where \[ A=\left[ \begin{array}{c@{\quad}c@{\quad}c} 2 & 1 & 2 \\ 4 & 0 & 1 \\ 1 & 3 & 0 \\ \end{array} \right] \quad \quad \quad B=\left[ \begin{array}{c@{\quad}c@{\quad}c} 3 & 0 & 5 \\ 1 & 2 & 1 \\ 0 & 3 & 1 \\ \end{array} \right]\!. \]
Let \({\bf a,b}\) be two vectors in the plane, \({\bf a}=(a_1,a_2), {\bf b} = (b_1, b_2)\), and let \(\lambda\) be a real number. Show that the area of the parallelogram determined by a and \({\bf b} + \lambda {\bf a}\) is the same as that determined by a and b. Sketch. Relate this result to a known property of determinants.
Find the volume of the parallelepiped determined by the vertices \((0,1,0),(1,1,1),\) \((0,2,0),\) \((3,1,2)\).
Given nonzero vectors a and b in \({\mathbb R}^3\), show that the vector \({\bf v} = \|{\bf a} \| {\bf b} + \|{\bf b} \| {\bf a}\) bisects the angle between a and b.
Show that the vectors \(\|\textbf{b}\|\textbf{a} +\|\textbf{a}\|\textbf{b}\) and \(\|\textbf{b}\|\textbf{a} -\|\textbf{a}\|\textbf{b}\) are orthogonal.
Use the triangle inequality to show that \( \| \textbf{v} - \textbf{w} \| \geq \ \Big| \| \textbf{v} \| - \| \textbf{w} \| \Big| \).
Use vector methods to prove that the distance from the point \((x_1,y_1)\) to the line \(ax+by\) \(=c\) is \[ \frac{|ax_1 + b y_1 -c|}{\sqrt{a^2 + b^2}}. \]
Verify that the direction of \({\bf b} \times {\bf c}\) is given by the right-hand rule, by choosing \({\bf b}, {\bf c}\) to be two of the vectors \({\bf i,j}\), and \({\bf k}\).
Show that two planes given by the equations \(Ax + By + Cz + D_1=0 \) and \(Ax + By + Cz + D_2 =0\) are parallel, and that the distance between them is \[ \frac{|D_1 - D_2|}{\sqrt{A^2 + B^2 + C^2}}. \]
Convert the following points from Cartesian to cylindrical and spherical coordinates and plot:
Convert the following points from cylindrical to Cartesian and spherical coordinates and plot:
Convert the following points from spherical to Cartesian and cylindrical coordinates and plot:
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Rewrite the equation \(z= x^2 - y^2\) using cylindrical and spherical coordinates.
Using spherical coordinates, show that \[ \phi = \cos^{-1} \Big( \frac{{\bf u \,{\cdot}\, k}}{\|{\bf u}\|} \Big), \] where \({\bf u} = x {\bf i} + y {\bf j} + z {\bf k}\). Interpret geometrically.
Verify the Cauchy–Schwarz and triangle inequalities for \[ {\bf x} = (3,2,1,0) \hbox{and} {\bf y} = (1,1,1,2). \]
Multiply the matrices \[ A = \Bigg[ \begin{array} &3 & 0 & 1 \\ 2 & 0 & 1 \\ 1 & 0 & 1 \end{array} \Bigg] \qquad \hbox{and } \qquad B = \Bigg[ \begin{array}{@{}c@{\quad}c@{\quad}c@{}} 1 & 0 & 1\\ 1 & 1 & 1\\ 0 & 0 & 1 \end{array} \Bigg]. \] Does \(AB =BA\)?
Find the volume of the parallelepiped spanned by the vectors \[ (1,0,1), (1,1,1), \hbox{and} (-3,2,0). \]
(For students with some knowledge of linear algebra.) Verify that a linear mapping \(T\) of \({\mathbb R}^n\) to \({\mathbb R}^n\) is determined by an \(n \times n\) matrix.
Find an equation for the plane that contains \((3,-1,2)\) and the line with equation \({\bf v} =(2,-1,0) + t(2,3,0)\).
The work \(W\) done in moving an object from (0, 0) to (7, 2) subject to a constant force \({\bf F}\) is \(W = {\bf F \,{\cdot}\, r}\), where \({\bf r}\) is the vector with its head at (7, 2) and tail at (0, 0). The units are feet and pounds.
If a particle with mass \(m\) moves with velocity \({\bf v}\), its momentum is \({\bf p} = m { \bf v}\). In a game of marbles, a marble with mass 2 grams (g) is shot with velocity 2 meters per second (m /s), hits two marbles with mass 1 g each, and comes to a dead halt. One of the marbles flies off with a velocity of 3 m/s at an angle of \(45^\circ\) to the incident direction of the larger marble, as in Figure 1.R.1. Assuming that the total momentum before and after the collision is the same (according to the law of conservation of momentum), at what angle and speed does the second marble move?
Show that for all \(x,y,z\), \[ \Bigg|\begin{array}{@{}c@{\quad}c@{\quad}c@{}} x+2 & y & z \\ z & y+1 & 10 \\ 5 & 5 & 2 \end{array} \Bigg| = - \Bigg| \begin{array}{@{}c@{\quad}c@{\quad}c@{}} y & x+2 & z \\ 1 & z-x-2 & 10-z \\ 5 & 5 & 2 \end{array} \Bigg| . \]
Show that \[ \Bigg| \begin{array}{@{}c@{\quad}c@{\quad}c@{}} 1 & x & x^2 \\ 1 & y & y^2 \\ 1 & z & z^2 \end{array} \Bigg| \ne 0 \] if \(x,y\), and \(z\) are all different.
Show that \[ \Bigg| \begin{array}{@{}r@{\quad}c@{\quad}r@{}} 66 & 628 & 246 \\ 88 & 435 & 24 \\ 2 & -1 & 1 \end{array} \Bigg| = \Bigg| \begin{array}{@{}r@{\quad}c@{\quad}r@{}} 68 & 627 & 247 \\ 86 & 436 & 23 \\ 2 & -1 & 1 \end{array} \Bigg|. \]
Show that \[ \Bigg| \begin{array}{@{}c@{\qquad}c@{\qquad}c@{}} n & n+1 & n+2 \\ n+3 & n+4 & n+5 \\ n+6 & n+ 7 & n+8 \end{array} \Bigg| \] has the same value no matter what \(n\) is. What is this value?
Are the following quantities vectors or scalars?
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Find a \(4 \times 4\) matrix \(C\) such that for every \(4 \times 4\) matrix \(A\) we have \(CA=3A\).
Let \[ A=\left[ \begin{array}{c@{\quad}c} 1 & 1 \\ 0 & 1 \\ \end{array} \right] \quad \quad \quad B=\left[ \begin{array}{c@{\quad}c} 1 & 0 \\ 2 & 1 \\ \end{array} \right] \]
Suppose \(\left[ \begin{array}{c@{\quad}c} a & b \\ c & d \\ \end{array} \right]\) is invertible and has integer entries.
What conditions must be satisfied for \(A=\left[ \begin{array}{c@{\quad}c} a & b \\ c & d \\ \end{array} \right]^{-1}\) to have integer entries?
The volume of a tetrahedron with concurrent edges a, b, c is given by \(V = \frac{1}{6} {\bf a} \,{\cdot}\, ( {\bf b} \times {\bf c}).\)
Use the following definition for Exercises 50 and \(51{:}\) Let \({\bf r}_1, \ldots ,{\bf r}_n\) be vectors in \({\mathbb R}^3\) from \(0\) to the masses \(m_1 , \ldots, m_n\). The center of mass is the vector \[ {\bf c} = \frac{\sum_{i\,{=}\,1}^n \,m_i {\bf r}_i}{\sum_{i\,{=}\,1}^n\, m_i}. \]
A tetrahedron sits in \(xyz\) coordinates with one vertex at \((0,0,0)\), and the three edges concurrent at \((0, 0, 0)\) are coincident with the vectors a, b, c.
Show that for any vector \({\bf r}\), the center of mass of a system satisfies \[ \sum_{i=1}^n \,m_i \|{\bf r} - {\bf r}_i\|^2 = \sum_{i=1}^n \,m_i \|{\bf r}_i - {\bf c}\|^2 + m \|{\bf r} - {\bf c} \|^2, \] where \(m = {\sum}_{i=1}^n \,m_i\) is the total mass of the system.
In Exercises 52 to 57, find a unit vector that has the given property.
Parallel to the line \(x=3t +1 , y=16t-2, z =- (t+2)\)
Orthogonal to the plane \(x-6y +z =12\)
Parallel to both the planes \(8x + y + z =1\) and \(x-y -z =0\)
Orthogonal to \({\bf i} + 2 {\bf j} - {\bf k}\) and to k
Orthogonal to the line \(x = 2t-1, y=-t-1, z=t+2\), and the vector \({\bf i}- {\bf j}\)
At an angle of \(30^\circ\) to i and making equal angles with j and k
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1For him, “acceptable” meant that the associative law of multiplication would hold.
2North British Review, 14 (1858), p. 57.
3Interestingly, if one is willing to continue to live with nonassociativity, there is also a vector product with most of the properties of the cross product in \({\mathbb R}^{7}\); this involves yet another number system called the octonians, which exists in \({\mathbb R}^{8}\). The nonexistence of a cross product in other dimensions is a result that goes beyond the scope of this text. For further information, see the American Mathematical Monthly, 74 (1967), pp. 188–194, and 90 (1983), p. 697, as well as J. Baez, “The Octonians,” Bulletin of the American Mathematical Society, 39 (2002), pp. 145–206. One can show that systems like the quaternions and octonians occur only in dimension 1 (the reals \({\mathbb R}\)), dimension 2 (the complex numbers), dimension 4 (the quaternions), and dimension 8 (the octonians). On the other hand, the “right” way to extend the cross product is to introduce the notion of differential forms, which exists in any dimension. We discuss their construction in Section 18.5.
4Sometimes called the Cauchy–Bunyakovskii–Schwarz inequality, or simply the CBS inequality, because it was independently discovered in special cases by the French mathematician Cauchy, the Russian mathematician Bunyakovskii, and the German mathematician Schwarz.
5To use a matrix \(A\) to get a mapping from vectors \({\bf x}=(x_1,\ldots ,x_n)\) to vectors \({\bf y}=(\kern1pty_1,\ldots ,y_n)\) according to the equation \(A{\bf x}^T={\bf y}^T\), we write the vectors in the column form \({\bf x}^T\) instead of the row form \((x_1,\ldots, x_n)\). This sudden switch from writing x as a row to writing x as a column is necessitated by standard conventions on matrix multiplication.
6In fact, Cramer’s rule from Section 11.3 provides one way to invert matrices. Numerically more efficient methods based on elimination methods are learned in linear algebra or computer science.