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# Solutions for Chapter 12: Vectors and the Geometry of Space

## Full solutions for Thomas' Calculus Early Transcendentals | 12th Edition

ISBN: 9780321588760

Solutions for Chapter 12: Vectors and the Geometry of Space

Solutions for Chapter 12
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##### ISBN: 9780321588760

Thomas' Calculus Early Transcendentals was written by and is associated to the ISBN: 9780321588760. This textbook survival guide was created for the textbook: Thomas' Calculus Early Transcendentals, edition: 12. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 12: Vectors and the Geometry of Space includes 76 full step-by-step solutions. Since 76 problems in chapter 12: Vectors and the Geometry of Space have been answered, more than 7322 students have viewed full step-by-step solutions from this chapter.

Key Calculus Terms and definitions covered in this textbook
• Arccotangent function

See Inverse cotangent function.

• Associative properties

a + (b + c) = (a + b) + c, a(bc) = (ab)c.

• Circular functions

Trigonometric functions when applied to real numbers are circular functions

• Complex conjugates

Complex numbers a + bi and a - bi

• Conjugate axis of a hyperbola

The line segment of length 2b that is perpendicular to the focal axis and has the center of the hyperbola as its midpoint

• De Moivreâ€™s theorem

(r(cos ? + i sin ?))n = r n (cos n? + i sin n?)

• Degree

Unit of measurement (represented by the symbol ) for angles or arcs, equal to 1/360 of a complete revolution

• Difference identity

An identity involving a trigonometric function of u - v

• Direction of an arrow

The angle the arrow makes with the positive x-axis

• Distance (in a coordinate plane)

The distance d(P, Q) between P(x, y) and Q(x, y) d(P, Q) = 2(x 1 - x 2)2 + (y1 - y2)2

• Domain of validity of an identity

The set of values of the variable for which both sides of the identity are defined

• Gaussian elimination

A method of solving a system of n linear equations in n unknowns.

• Polar distance formula

The distance between the points with polar coordinates (r1, ?1 ) and (r2, ?2 ) = 2r 12 + r 22 - 2r1r2 cos 1?1 - ?22

An equation that can be written in the form ax 2 + bx + c = 01a ? 02

• Root of a number

See Principal nth root.

• Semiperimeter of a triangle

One-half of the sum of the lengths of the sides of a triangle.

• Tangent

The function y = tan x

• Transformation

A function that maps real numbers to real numbers.

• Vector equation for a line in space

The line through P0(x 0, y0, z0) in the direction of the nonzero vector V = <a, b, c> has vector equation r = r0 + tv , where r = <x,y,z>.

• Zero factorial

See n factorial.

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