 7.2.1: In 16, graph the given point. Use the same coordinate axes.
 7.2.2: In 16, graph the given point. Use the same coordinate axes.
 7.2.3: In 16, graph the given point. Use the same coordinate axes.
 7.2.4: In 16, graph the given point. Use the same coordinate axes.
 7.2.5: In 16, graph the given point. Use the same coordinate axes.
 7.2.6: In 16, graph the given point. Use the same coordinate axes.
 7.2.7: In 710, describe geometrically all points P(x, y, z) that satisfy t...
 7.2.8: In 710, describe geometrically all points P(x, y, z) that satisfy t...
 7.2.9: In 710, describe geometrically all points P(x, y, z) that satisfy t...
 7.2.10: In 710, describe geometrically all points P(x, y, z) that satisfy t...
 7.2.11: Give the coordinates of the vertices of the rectangular parallelepi...
 7.2.12: In FIGURE 7.2.9, two vertices are shown of a rectangular parallelep...
 7.2.13: Consider the point P(2, 5, 4). (a) If lines are drawn from P perpen...
 7.2.14: Determine an equation of a plane parallel to a coordinate plane tha...
 7.2.15: In 1520, describe the locus of points P(x, y, z) that satisfy the g...
 7.2.16: In 1520, describe the locus of points P(x, y, z) that satisfy the g...
 7.2.17: In 1520, describe the locus of points P(x, y, z) that satisfy the g...
 7.2.18: In 1520, describe the locus of points P(x, y, z) that satisfy the g...
 7.2.19: In 1520, describe the locus of points P(x, y, z) that satisfy the g...
 7.2.20: In 1520, describe the locus of points P(x, y, z) that satisfy the g...
 7.2.21: In 21 and 22, find the distance between the given points.
 7.2.22: In 21 and 22, find the distance between the given points.
 7.2.23: Find the distance from the point (7, 3, 4) to (a) the yzplane and ...
 7.2.24: Find the distance from the point (6, 2, 3) to (a) the xzplane and ...
 7.2.25: In 2528, the given three points form a triangle. Determine which tr...
 7.2.26: In 2528, the given three points form a triangle. Determine which tr...
 7.2.27: In 2528, the given three points form a triangle. Determine which tr...
 7.2.28: In 2528, the given three points form a triangle. Determine which tr...
 7.2.29: In 29 and 30, use the distance formula to prove that the given poin...
 7.2.30: In 29 and 30, use the distance formula to prove that the given poin...
 7.2.31: In 31 and 32, solve for the unknown.
 7.2.32: In 31 and 32, solve for the unknown.
 7.2.33: In 33 and 34, find the coordinates of the midpoint of the line segm...
 7.2.34: In 33 and 34, find the coordinates of the midpoint of the line segm...
 7.2.35: The coordinates of the midpoint of the line segment between P1(x1, ...
 7.2.36: Let P3 be the midpoint of the line segment between P1(3, 4, 1) and ...
 7.2.37: In 3740, find the vector P1P2 ! .
 7.2.38: In 3740, find the vector P1P2 ! .
 7.2.39: In 3740, find the vector P1P2 ! .
 7.2.40: In 3740, find the vector P1P2 ! .
 7.2.41: In 4148, a 1, 3, 2, b 1, 1, 1, and c 2, 6, 9. Find the indicated ve...
 7.2.42: In 4148, a 1, 3, 2, b 1, 1, 1, and c 2, 6, 9. Find the indicated ve...
 7.2.43: In 4148, a 1, 3, 2, b 1, 1, 1, and c 2, 6, 9. Find the indicated ve...
 7.2.44: In 4148, a 1, 3, 2, b 1, 1, 1, and c 2, 6, 9. Find the indicated ve...
 7.2.45: In 4148, a 1, 3, 2, b 1, 1, 1, and c 2, 6, 9. Find the indicated ve...
 7.2.46: In 4148, a 1, 3, 2, b 1, 1, 1, and c 2, 6, 9. Find the indicated ve...
 7.2.47: In 4148, a 1, 3, 2, b 1, 1, 1, and c 2, 6, 9. Find the indicated ve...
 7.2.48: In 4148, a 1, 3, 2, b 1, 1, 1, and c 2, 6, 9. Find the indicated ve...
 7.2.49: Find a unit vector in the opposite direction of a 10, 5, 10.
 7.2.50: Find a unit vector in the same direction as a i 3j 2k.
 7.2.51: Find a vector b that is four times as long as a i j k in the same d...
 7.2.52: Find a vector b for which ibi 1 2 that is parallel to a 6, 3, 2 but...
 7.2.53: Using the vectors a and b shown in FIGURE 7.2.10, sketch the averag...
Solutions for Chapter 7.2: Vectors in 3Space
Full solutions for Advanced Engineering Mathematics  6th Edition
ISBN: 9781284105902
Solutions for Chapter 7.2: Vectors in 3Space
Get Full SolutionsThis textbook survival guide was created for the textbook: Advanced Engineering Mathematics , edition: 6. Since 53 problems in chapter 7.2: Vectors in 3Space have been answered, more than 39675 students have viewed full stepbystep solutions from this chapter. Chapter 7.2: Vectors in 3Space includes 53 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9781284105902.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.