Book cover for Advanced Engineering Mathematics

Advanced Engineering Mathematics

Dennis G. Zill, Michael R. Cullen

ISBN #9780763740955

3rd Edition

4,310 Questions

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17,647 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section emphasizes the core operations of vector algebra and matrix manipulation, including addition, scalar multiplication, dot and cross products, and applications in geometry. It also introduces the concept of vector spaces and subspaces, along with methods to achieve orthogonal and orthonormal bases via the Gram–Schmidt process. Mastery of these topics is foundational for advanced studies in linear algebra, calculus, and applied mathematics.

Learning Objectives

1

Understand and perform basic vector operations including addition, scalar multiplication, dot product, and cross product.

2

Apply matrix algebra, including matrix addition, subtraction, and multiplication, to solve systems of equations.

3

Develop proficiency in the concepts of vector spaces, subspaces, linear independence, and orthonormal bases.

4

Execute the Gram–Schmidt orthogonalization process to convert a given basis into an orthogonal (or orthonormal) basis.

5

Interpret and solve applied problems involving vectors and matrices in geometric contexts such as lines, planes, and spheres.

Key Concepts

CONCEPT

DEFINITION

Vector

A quantity that has both magnitude and direction, represented in coordinate form (e.g., ⟨x, y, z⟩).

Matrix

A rectangular array of numbers arranged in rows and columns, used to represent linear transformations and solve systems of equations.

Dot Product

An operation that multiplies corresponding entries of two vectors and sums the results; it gives a scalar and can be used to determine angles between vectors.

Cross Product

A binary operation on two vectors in three-dimensional space which results in a vector perpendicular to both original vectors, with magnitude equal to the area of the parallelogram they span.

Vector Space

A collection of objects (vectors) that can be added together and multiplied by scalars, satisfying a set of axioms (closure, associativity, distributivity, etc.).

Subspace

A subset of a vector space that is itself a vector space under the same operations.

Linear Independence

A set of vectors is said to be linearly independent if no vector in the set can be written as a linear combination of the others.

Gram–Schmidt Process

A method for converting a set of linearly independent vectors into an orthogonal (or orthonormal) set spanning the same subspace.

Example Problems

Example 1

(a) $6 \mathbf{i}+12 \mathbf{j}$ (b) $\mathbf{i}+8 \mathbf{j}$ (c) $3 \mathbf{i}$ (d) $\sqrt{65}$ (e) 3

Example 2

(a) $\langle 3,3\rangle$ (b) $\langle 3,4\rangle$ (c) $\langle-1,-2\rangle$ (d) $5$ (e) $\sqrt{5}$

Example 3

(a) $\langle 12,0\rangle$ (b) $\langle 4,-5\rangle$ (c) $\langle 4,5\rangle$ (d) $\sqrt{41}$ (e) $\sqrt{41}$

Example 4

(a) $\frac{1}{2} \mathbf{i}-\frac{1}{2} \mathbf{j}$ (b) $\frac{2}{3} \mathbf{i}+\frac{2}{3} \mathbf{j}$ (c) $-\frac{1}{3} i-j$ (d) $2 \sqrt{2} / 3$ (e) $\sqrt{10} / 3$

Example 5

(a) $-9 \mathbf{i}+6 \mathbf{j}$ (b) $-3 \mathbf{i}+9 \mathbf{j}$ (c) $-3 \mathbf{i}-5 \mathbf{j}$ (d) $3 \sqrt{10}$ (e) $\sqrt{34}$
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Step-by-Step Explanations

QUESTION

Given vectors a = ⟨2, −3, 4⟩ and b = ⟨−1, 5, 2⟩, find a · b.

STEP-BY-STEP ANSWER:

Step 1: Multiply the corresponding components: 2×(−1), (−3)×5, 4×2.
Step 2: Compute these products: -2, -15, 8.
Step 3: Sum the products: -2 + (-15) + 8 = -9.
Final Answer: a · b = -9.

Dot Product Example

QUESTION

Given vectors a = ⟨1, 2, 3⟩ and b = ⟨4, 5, 6⟩, compute a × b.

STEP-BY-STEP ANSWER:

Step 1: Write down the determinant formula: a × b = ⟨a2b3 – a3b2, a3b1 – a1b3, a1b2 – a2b1⟩.
Step 2: Substitute values: a2b3 = 2×6 = 12, a3b2 = 3×5 = 15, a3b1 = 3×4 = 12, a1b3 = 1×6 = 6, a1b2 = 1×5 = 5, a2b1 = 2×4 = 8.
Step 3: Compute components: first = 12 − 15 = -3; second = 12 − 6 = 6; third = 5 − 8 = -3.
Final Answer: a × b = ⟨-3, 6, -3⟩.

Cross Product Example

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Common Mistakes

  • Mixing up the order of terms in the cross product, which is anti-commutative (i.e., a × b = ?(b × a)).
  • Forgetting that the dot product results in a scalar, not a vector.
  • Errors in matching matrix dimensions when adding or multiplying matrices.
  • Overlooking the requirements of vector space axioms when determining if a set is a subspace.
  • Misapplying the Gram–Schmidt process by not subtracting the correct projections, which leads to non-orthogonal vectors.