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Chapter 11: Problem 10

Explain why each of the following expressions makes sense. \(\begin{array}{ll}{\text { (a) }(\mathbf{u} \cdot \mathbf{v}) \mathbf{w}} & {\text { (b) }(\mathbf{u} \cdot \mathbf{v})(\mathbf{v} \cdot \mathbf{w})} \\\ {\text { (c) } \mathbf{u} \cdot \mathbf{v}+k} & {\text { (d) }(k \mathbf{u}) \cdot \mathbf{v}}\end{array}\)

Short Answer

Expert verified
Each expression combines vectors and scalars in mathematically valid operations.

Step by step solution

01

Understanding Dot Products

The dot product \( \mathbf{u} \cdot \mathbf{v} \) of two vectors \( \mathbf{u} \) and \( \mathbf{v} \) results in a scalar (a single number). This is vital for understanding the operations (a) through (d) as operations must be mathematically valid, including combinations of vectors and scalars.
02

Analyzing Expression (a)

In (a), \((\mathbf{u} \cdot \mathbf{v}) \mathbf{w}\), the dot product \(\mathbf{u} \cdot \mathbf{v}\) is a scalar, which can then multiply vector \(\mathbf{w}\). Multiplying a scalar by a vector is a valid operation, resulting in a new vector. Thus, (a) makes sense mathematically.
03

Analyzing Expression (b)

For (b), \((\mathbf{u} \cdot \mathbf{v})(\mathbf{v} \cdot \mathbf{w})\), both \(\mathbf{u} \cdot \mathbf{v}\) and \(\mathbf{v} \cdot \mathbf{w}\) are scalars. Multiplying two scalars together is a valid mathematical operation, resulting in another scalar. Therefore, (b) is a sensible expression.
04

Analyzing Expression (c)

In (c), \(\mathbf{u} \cdot \mathbf{v} + k\), \(\mathbf{u} \cdot \mathbf{v}\) is a scalar resulting from the dot product and \(k\) is a given scalar. Adding two scalars is a fundamental arithmetic operation, so (c) is a valid expression.
05

Analyzing Expression (d)

In (d), \((k \mathbf{u}) \cdot \mathbf{v}\), multiplying vector \(\mathbf{u}\) by a scalar \(k\) yields a new vector. The dot product of this vector \((k \mathbf{u})\) with \(\mathbf{v}\) gives a scalar. Since each operation鈥攕calar multiplication followed by a dot product鈥攊s defined, (d) is valid.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Scalar Multiplication
Scalar multiplication is a fundamental aspect of vector operations. It involves multiplying a vector by a scalar, which is simply a real number. This operation scales the vector, affecting its magnitude and possibly its direction, depending on the scalar's sign. For example, if you have a vector \( \mathbf{u} \) and a scalar \( k \), the product \( k \mathbf{u} \) is obtained by multiplying each component of \( \mathbf{u} \) by \( k \).
This process has the following characteristics:
  • If \( k > 0 \), the vector maintains its direction, only stretching or compressing.
  • If \( k < 0 \), the vector not only stretches or compresses but also reverses its direction.
  • If \( k = 0 \), the result is the zero vector, which has no direction.
Scalar multiplication is crucial because it shows how vectors can be altered in a straightforward manner, and it plays a critical role in operations such as the dot product.
Dot Product
The dot product, also known as the scalar product, is a way of multiplying two vectors that results in a scalar. To compute the dot product of two vectors \( \mathbf{u} \) and \( \mathbf{v} \), you multiply corresponding components of the vectors and then add them together. It is often represented as \( \mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + \ldots + u_nv_n \).
Here are some properties of the dot product:
  • The result is always a scalar, not a vector.
  • It can tell you whether two vectors are orthogonal; if \( \mathbf{u} \cdot \mathbf{v} = 0 \), they are perpendicular.
  • It relates to the cosine of the angle between the vectors: \( \mathbf{u} \cdot \mathbf{v} = ||\mathbf{u}|| ||\mathbf{v}|| \cos \theta \).
The dot product is extensively used in physics and engineering to project vectors and determine angles between them.
Valid Mathematical Operations
In vector mathematics, ensuring operations are valid is key to solving problems correctly. Valid operations include combining vectors and scalars in ways that adhere to mathematical rules. The essence is observing the structure of vector and scalar operations, like those seen in the exercise expressions. Let's consider a few points:
  • Multiplying a scalar and a vector (scalar multiplication) is valid and results in a vector.
  • Performing a dot product between two vectors results in a scalar.
  • Adding or multiplying scalars follows basic arithmetic rules.
All these ensure the expression remains mathematically sound, maintaining the integrity of calculations in physics, engineering, and beyond.
Vectors and Scalars
Vectors and scalars are core elements in mathematics, especially in fields like physics and engineering. Vectors are quantities with both magnitude and direction, such as velocity or force. They are typically represented in a coordinate system as arrows.
Scalars, on the other hand, are quantities with only magnitude. Examples include time, temperature, and distance. Understanding the distinction:
  • Vectors: Represented as \( \mathbf{v} = \langle v_1, v_2, \ldots, v_n \rangle \).
  • Scalars: Simple numbers like \( k, 5, -3 \).
  • Vectors can be added and subtracted as long as they have the same dimensions.
  • Vectors are multiplied by scalars to scale their influence.
This concept is vital for implementing and understanding operations like dot products and scalar multiplication, which manipulate these elements according to their characteristics.

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