 1.1: Let v = 3i + 4j + 5k and w = i j + k. Compute v + w, 3v, 6v + 8w, 2...
 1.2: Repeat Exercise 1 with v = 2j + k and w = i k
 1.3: (a) Find the equation of the line through (1, 2, 1) in the directio...
 1.4: (a) Find the equation of the line through (0, 1, 0) in the directio...
 1.5: Find an equation for the plane containing the points (2, 1, 1), (3,...
 1.6: Find an equation for a line that is parallel to the plane 2x 3y + 5...
 1.7: Compute v w for the following sets of vectors: (a) v = i + j; w = k...
 1.8: Compute v w for the vectors in Exercise 7. [Only part (b) is solved...
 1.9: Find the cosine of the angle between the vectors in Exercise 7. [On...
 1.10: Find the area of the parallelogram spanned by the vectors in Exerci...
 1.11: Use vector notation to describe the triangle in space whose vertice...
 1.12: Show that three vectors a, b, c lie in the same plane through the o...
 1.13: For real numbers a1, a2, a3, b1, b2, b3, show that (a1b1 + a2b2 + a...
 1.14: Let u, v, w be unit vectors that are orthogonal to each other. If a...
 1.15: Find the products AB and B A where A = 152 023 102 B = 201 130 241
 1.16: Find the products AB and B A where A = 212 401 130 B = 305 121 031 .
 1.17: Let a, b be two vectors in the plane, a = (a1, a2), b = (b1, b2), a...
 1.18: Find the volume of the parallelepiped determined by the vertices (0...
 1.19: Given nonzero vectors a and b in R3, show that the vector v = a b +...
 1.20: Show that the vectors b a + a b and b a a b are orthogonal.
 1.21: Use the triangle inequality to show that v w v w .
 1.22: Use vector methods to prove that the distance from the point (x1, y...
 1.23: Verify that the direction of b c is given by the righthand rule, b...
 1.24: (a) Suppose a b = a b for all b. Show that a = a . (b) Suppose a b ...
 1.25: (a) Using vector methods, show that the distance between two nonpar...
 1.26: Show that two planes given by the equations Ax + By + Cz + D1 = 0 a...
 1.27: (a) Prove that the area of the triangle in the plane with vertices ...
 1.28: Convert the following points from Cartesian to cylindrical and sphe...
 1.29: Convert the following points from cylindrical to Cartesian and sphe...
 1.30: Convert the following points from spherical to Cartesian and cylind...
 1.31: Rewrite the equation z = x y using cylindrical and spherical coordi...
 1.32: Using spherical coordinates, show that = cos1 u k u , where u = xi ...
 1.33: Verify the CauchySchwarz and triangle inequalities for x = (3, 2, 1...
 1.34: Multiply the matrices A = 301 201 101 and B = 101 111 001 . Does AB...
 1.35: (a) Show that for two n n matrices A and B, and x Rn, (AB)x = A(Bx)...
 1.36: Find the volume of the parallelepiped spanned by the vectors (1, 0,...
 1.37: (For students with some knowledge of linear algebra.) Verify that a...
 1.38: Find an equation for the plane that contains (3, 1, 2) and the line...
 1.39: The work W done in moving an object from (0, 0) to (7, 2) subject t...
 1.40: If a particle with mass m moves with velocity v, its momentum is p ...
 1.41: Show that for all x, y, z, x + 2 y z z y + 1 10 5 52 = y x + 2 z 1 ...
 1.42: Show that 1 x x2 1 y y2 1 z z2 = 0 if x, y, and z are all different.
 1.43: Show that 66 628 246 88 435 24 2 1 1 = 68 627 247 86 436 23 2 1 1
 1.44: Show that n n + 1 n + 2 n + 3 n + 4 n + 5 n + 6 n + 7 n + 8 has the...
 1.45: Are the following quantities vectors or scalars? (a) The current po...
 1.46: Find a 4 4 matrix C such that for every 4 4 matrix A we have C A = 3A.
 1.47: Let A = 1 1 0 1 B = 1 0 2 1 (a) Find A1, B1, and (AB)1. (b) Show th...
 1.48: Suppose a b c d is invertible and has integer entries. What conditi...
 1.49: The volume of a tetrahedron with concurrent edges a, b, c is given ...
 1.50: Use the following definition for Exercises 50 and 51: Let r1, . . ....
 1.51: Use the following definition for Exercises 50 and 51: Let r1, . . ....
 1.52: In Exercises 52 to 57, find a unit vector that has the given proper...
 1.53: In Exercises 52 to 57, find a unit vector that has the given proper...
 1.54: In Exercises 52 to 57, find a unit vector that has the given proper...
 1.55: In Exercises 52 to 57, find a unit vector that has the given proper...
 1.56: In Exercises 52 to 57, find a unit vector that has the given proper...
 1.57: In Exercises 52 to 57, find a unit vector that has the given proper...
Solutions for Chapter 1: The Geometry of Euclidean Space
Full solutions for Vector Calculus  6th Edition
ISBN: 9781429215084
Solutions for Chapter 1: The Geometry of Euclidean Space
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Vector Calculus was written by and is associated to the ISBN: 9781429215084. Since 57 problems in chapter 1: The Geometry of Euclidean Space have been answered, more than 3080 students have viewed full stepbystep solutions from this chapter. Chapter 1: The Geometry of Euclidean Space includes 57 full stepbystep solutions. This textbook survival guide was created for the textbook: Vector Calculus, edition: 6.

Causation
A relationship between two variables in which the values of the response variable are directly affected by the values of the explanatory variable

Common logarithm
A logarithm with base 10.

Compounded annually
See Compounded k times per year.

Coordinate plane
See Cartesian coordinate system.

Directed angle
See Polar coordinates.

Equilibrium point
A point where the supply curve and demand curve intersect. The corresponding price is the equilibrium price.

Limit at infinity
limx: qƒ1x2 = L means that ƒ1x2 gets arbitrarily close to L as x gets arbitrarily large; lim x: q ƒ1x2 means that gets arbitrarily close to L as gets arbitrarily large

Linear regression line
The line for which the sum of the squares of the residuals is the smallest possible

Order of an m x n matrix
The order of an m x n matrix is m x n.

Pascal’s triangle
A number pattern in which row n (beginning with n = 02) consists of the coefficients of the expanded form of (a+b)n.

Probability function
A function P that assigns a real number to each outcome O in a sample space satisfying: 0 … P1O2 … 1, P12 = 0, and the sum of the probabilities of all outcomes is 1.

Quotient identities
tan ?= sin ?cos ?and cot ?= cos ? sin ?

Reciprocal of a real number
See Multiplicative inverse of a real number.

Reflection through the origin
x, y and (x,y) are reflections of each other through the origin.

Response variable
A variable that is affected by an explanatory variable.

Scatter plot
A plot of all the ordered pairs of a twovariable data set on a coordinate plane.

Solution of a system in two variables
An ordered pair of real numbers that satisfies all of the equations or inequalities in the system

Standard form of a polynomial function
ƒ(x) = an x n + an1x n1 + Á + a1x + a0

Work
The product of a force applied to an object over a given distance W = ƒFƒ ƒAB!ƒ.

zaxis
Usually the third dimension in Cartesian space.