 12.3.1: Which of the following expressions are meaningful? Which are meanin...
 12.3.2: Find a b 2, 3 b 0.7, 1.2
 12.3.3: Find a ba 2, b 5, 12 1
 12.3.4: Find a ba 6, 2, 3 b 2, 5, 1
 12.3.5: Find a ba 4, 1, b 6, 3, 8 1
 12.3.6: Find a ba p, p, 2p b 2q, q, q
 12.3.7: Find a ba 2i j b i j k
 12.3.8: Find a ba 3i 2j k b 4i 5k
 12.3.9: Find a ba 6 b 5 a b 2
 12.3.10: Find a ba 3 b s6 a b 45
 12.3.11: If u is a unit vector, find and .
 12.3.12: If u is a unit vector, find and .
 12.3.13: (a) Show that . (b) Show that .
 12.3.14: A street vendor sells hamburgers, hot dogs, and soft drinks on a gi...
 12.3.15: Find the angle between the vectors. (First find an exact expression...
 12.3.16: Find the angle between the vectors. (First find an exact expression...
 12.3.17: Find the angle between the vectors. (First find an exact expression...
 12.3.18: Find the angle between the vectors. (First find an exact expression...
 12.3.19: Find the angle between the vectors. (First find an exact expression...
 12.3.20: Find the angle between the vectors. (First find an exact expression...
 12.3.21: Find, correct to the nearest degree, the three angles of the triang...
 12.3.22: Find, correct to the nearest degree, the three angles of the triang...
 12.3.23: Determine whether the given vectors are orthogonal, parallel, or ne...
 12.3.24: Determine whether the given vectors are orthogonal, parallel, or ne...
 12.3.25: Use vectors to decide whether the triangle with vertices , , and is...
 12.3.26: Find the values of such that the angle between the vectors , and is .
 12.3.27: Find a unit vector that is orthogonal to both and .
 12.3.28: Find two unit vectors that make an angle of with .
 12.3.29: Find the acute angle between the lines. 2x y 3 3x y 7
 12.3.30: Find the acute angle between the lines. x 2y 7 5x y 2
 12.3.31: Find the acute angles between the curves at their points of interse...
 12.3.32: Find the acute angles between the curves at their points of interse...
 12.3.33: Find the direction cosines and direction angles of the vector. (Giv...
 12.3.34: Find the direction cosines and direction angles of the vector. (Giv...
 12.3.35: Find the direction cosines and direction angles of the vector. (Giv...
 12.3.36: Find the direction cosines and direction angles of the vector. (Giv...
 12.3.37: Find the direction cosines and direction angles of the vector. (Giv...
 12.3.38: If a vector has direction angles and , find the third direction ang...
 12.3.39: Find the scalar and vector projections of onto . a 5, 12 b 4, 6
 12.3.40: Find the scalar and vector projections of onto . a 1, 4 b 2, 3
 12.3.41: Find the scalar and vector projections of onto . a 3, 6, 2 b 1, 2, 3
 12.3.42: Find the scalar and vector projections of onto . a 2, 3, 6 b 5, 1, 4
 12.3.43: Find the scalar and vector projections of onto . 2a 2i j 4k k
 12.3.44: Find the scalar and vector projections of onto . a i j k b i j k
 12.3.45: Show that the vector is orthogonal to . (It is called an orthogonal...
 12.3.46: For the vectors in Exercise 40, find and illustrate by drawing the ...
 12.3.47: If , find a vector such that .
 12.3.48: Suppose that and are nonzero vectors. (a) Under what circumstances ...
 12.3.49: Find the work done by a force that moves an object from the point t...
 12.3.50: A tow truck drags a stalled car along a road. The chain makes an an...
 12.3.51: A sled is pulled along a level path through snow by a rope. A 30lb...
 12.3.52: A boat sails south with the help of a wind blowing in the direction...
 12.3.53: Use a scalar projection to show that the distance from a point to t...
 12.3.54: If , and , show that the vector equation represents a sphere, and f...
 12.3.55: Find the angle between a diagonal of a cube and one of its edges.
 12.3.56: Find the angle between a diagonal of a cube and a diagonal of one o...
 12.3.57: A molecule of methane, , is structured with the four hydrogen atoms...
 12.3.58: If , where , , and are all nonzero vectors, show that bisects the a...
 12.3.59: Prove Properties 2, 4, and 5 of the dot product (Theorem 2).
 12.3.60: Suppose that all sides of a quadrilateral are equal in length and o...
 12.3.61: Use Theorem 3 to prove the CauchySchwarz Inequality:
 12.3.62: The Triangle Inequality for vectors is (a) Give a geometric interpr...
 12.3.63: The Parallelogram Law states that (a) Give a geometric interpretati...
 12.3.64: Show that if and are orthogonal, then the vectors and must have the...
Solutions for Chapter 12.3: THE DOT PRODUCT
Full solutions for Multivariable Calculus,  7th Edition
ISBN: 9780538497879
Solutions for Chapter 12.3: THE DOT PRODUCT
Get Full SolutionsSince 64 problems in chapter 12.3: THE DOT PRODUCT have been answered, more than 21974 students have viewed full stepbystep solutions from this chapter. Multivariable Calculus, was written by and is associated to the ISBN: 9780538497879. This textbook survival guide was created for the textbook: Multivariable Calculus,, edition: 7. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 12.3: THE DOT PRODUCT includes 64 full stepbystep solutions.

Acute triangle
A triangle in which all angles measure less than 90°

Additive inverse of a complex number
The opposite of a + bi, or a  bi

Base
See Exponential function, Logarithmic function, nth power of a.

Characteristic polynomial of a square matrix A
det(xIn  A), where A is an n x n matrix

End behavior
The behavior of a graph of a function as.

Equilibrium price
See Equilibrium point.

Factored form
The left side of u(v + w) = uv + uw.

Imaginary axis
See Complex plane.

Infinite sequence
A function whose domain is the set of all natural numbers.

Inverse properties
a + 1a2 = 0, a # 1a

Mapping
A function viewed as a mapping of the elements of the domain onto the elements of the range

Open interval
An interval that does not include its endpoints.

Plane in Cartesian space
The graph of Ax + By + Cz + D = 0, where A, B, and C are not all zero.

Positive numbers
Real numbers shown to the right of the origin on a number line.

Quadratic formula
The formula x = b 2b2  4ac2a used to solve ax 2 + bx + c = 0.

Real axis
See Complex plane.

Row echelon form
A matrix in which rows consisting of all 0’s occur only at the bottom of the matrix, the first nonzero entry in any row with nonzero entries is 1, and the leading 1’s move to the right as we move down the rows.

Standard position (angle)
An angle positioned on a rectangular coordinate system with its vertex at the origin and its initial side on the positive xaxis

Variance
The square of the standard deviation.

zaxis
Usually the third dimension in Cartesian space.