There are three ways in which vectors can be multiplied, but none is exactly like the usual algebraic multiplication. As you read this material, keep in mind that a vector-capable calculator will help you multiply vectors only if you understand the basic rules of that multiplication.
Multiplying a Vector by a Scalar
If we multiply a vector by a scalar s, we get a new vector. Its magnitude is the product of the magnitude of and the absolute value of s. Its direction is the direction of if s is positive but the opposite direction if s is negative. To divide by s, we multiply by 1/s.
Multiplying a Vector by a Vector
There are two ways to multiply a vector by a vector: one way produces a scalar (called the scalar product), and the other produces a new vector (called the vector product). (Students commonly confuse the two ways.)
The Scalar Product
The scalar product of the vectors and in Fig. 3-18a is written as and defined to be
(3-20)
where a is the magnitude of
, b is the magnitude of
ath>
Note that there are only scalars on the right side of Eq. 3-20 (including the value of cos θ). Thus ⋅ on the left side represents a scalar quantity. Because of the notation, ⋅ is also known as the dot product and is spoken as "a dot b."
A dot product can be regarded as the product of two quantities:
(1) the magnitude of one of the vectors and
(2) the scalar component of the second vector along the direction of the first vector.
For example, in Fig. 3-18b,
has a scalar component a cos θ along the direction of
;
note that a perpendicular dropped from the head of
onto
determines that component. Similarly,
math>
If the angle between two vectors is 0o, the component of one vector along the other is maximum, and so also is the dot product of the vectors. If, instead, θ is 90°, the component of one vector along the other is zero, and so is the dot product.
Equation 3-20 can be rewritten as follows to emphasize the components:
(3-21)
The commutative law applies to a scalar product, so we can write
When two vectors are in unit-vector notation, we write their dot product as
which we can expand according to the distributive law: Each vector component
of the first vector is to be dotted with each vector component of the second vector.
By doing so, we can show that
Figure 3-18 (a) Two vectors
The Vector Product
The vector product of
C = ab sin θ (3-24)
where θ is the smaller of the two angles between
• If
The direction of
The order of the vector multiplication is important. In Fig. 3-19b, we are
determining the direction of
In other words, the commutative law does not apply to a vector product. In unit-vector notation, we write
which can be expanded according to the distributive law; that is, each component of the first vector is to be crossed with each component of the second vector. For example, in the expansion of Eq. 3-26, we have
In the last step we used Eq. 3-24 to evaluate the magnitude of
n the last step we used Eq. 3-24 to evaluate the magnitude of ' as unity.
(These vectors and each have a magnitude of unity, and the angle between
Continuing to expand Eq. 3-26, you can show that
A determinant or a vector-capable calculator can also be used.
To check whether any xyz coordinate system is a right-handed coordinate
system, use the right-hand rule for the cross product
Figure 3-19 Illustration of the right-hand rule for vector products. (a) Sweep vector
About the Authors
David Halliday was an American physicist known for his physics textbooks, Physics and Fundamentals of Physics, which he wrote with Robert Resnick. Both textbooks have been in continuous use since 1960 and are available in more than 47 languages.
Robert Resnick was a physics educator and author of physics textbooks. He was born in Baltimore, Maryland on January 11, 1923 and graduated from the Baltimore City College high school in 1939. He received his B.A. in 1943 and his Ph.D. in 1949, both in physics from Johns Hopkins University.
The 10th edition of Halliday's Fundamentals of Physics, Extended building upon previous issues by offering several new features and additions. The new edition offers most accurate, extensive and varied set of assessment questions of any course management program in addition to all questions including some form of question assistance including answer specific feedback to facilitate success. The text also offers multimedia presentations (videos and animations) of much of the material that provide an alternative pathway through the material for those who struggle with reading scientific exposition.
Furthermore, the book includes math review content in both a self-study module for more in-depth review and also in just-in-time math videos for a quick refresher on a specific topic. The Halliday content is widely accepted as clear, correct, and complete. The end-of-chapters problems are without peer. The new design, which was introduced in 9e continues with 10e, making this new edition of Halliday the most accessible and reader-friendly book on the market.
A Reader says,"As many reviewers have noted, this is a great physics book used widely in university technical programs as a first course in technical physics, with calculus. I find it is the one book I start with when trying to understand physical concepts at a useful but basic level. It has broad coverage and is well written . To go beyond this book requires specialized books on each topic of interest (electromagnetics, quantum mechanics, thermodynamics, etc.)."
Reader Frank says, "The treatment is sound, thorough, and clear. I've owned the early editions of Halliday and Resnick for years. I'm very happy that I updated my library with this 10th edition. The topics are covered in a very logical order. The study features and worked examples are outstanding. Don't hesitate to buy this book! Reading it is awesome on the Kindle app on the iPad."
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