Problem 43 - Calculus Early Transcendentals: Pearson New International Edition

Question

Which of the following do not make sense? (a) \(\mathbf{u} \cdot(\mathbf{v} \cdot \mathbf{w})\) (b) \((\mathbf{u} \cdot \mathbf{w})+\mathbf{w}\) (c) \(\|\mathbf{u}\|(\mathbf{v} \cdot \mathbf{w})\) (d) \((\mathbf{u} \cdot \mathbf{v}) \mathbf{w}\)

Step-by-Step Solution

Answer

(a) and (b) do not make sense.

Step 1: Understanding Dot Products and Vectors

The dot product \( \mathbf{a} \cdot \mathbf{b} \) results in a scalar. Scalars are just regular numbers, not vectors.

Step 2: Analyze Expression (a)

The expression \( \mathbf{u} \cdot (\mathbf{v} \cdot \mathbf{w}) \) suggests taking the dot product of three vectors. Since \( (\mathbf{v} \cdot \mathbf{w}) \) is a scalar, the expression becomes \( \mathbf{u} \cdot \text{(scalar)} \) which is not defined in vector algebra. Therefore, expression (a) does not make sense.

Step 3: Analyze Expression (b)

The expression \( (\mathbf{u} \cdot \mathbf{w}) + \mathbf{w} \) involves adding a scalar (result of the dot product \( \mathbf{u} \cdot \mathbf{w} \)) to a vector \( \mathbf{w} \). Adding a scalar to a vector is not a valid operation. Thus, (b) does not make sense.

Step 4: Analyze Expression (c)

The expression \( \| \mathbf{u} \| (\mathbf{v} \cdot \mathbf{w}) \) involves multiplying two scalars: \( \| \mathbf{u} \| \) (magnitude of a vector, a scalar) and \( \mathbf{v} \cdot \mathbf{w} \) (a scalar). This results in a scalar, and the expression makes sense.

Step 5: Analyze Expression (d)

The expression \( (\mathbf{u} \cdot \mathbf{v}) \mathbf{w} \) involves multiplying a scalar \( (\mathbf{u} \cdot \mathbf{v}) \) by a vector \( \mathbf{w} \). This is a valid operation, which scales the vector by the scalar. Thus, (d) makes sense.

Key Concepts

Dot Product

The dot product, also known as the scalar product, is a fundamental operation in vector algebra. It combines two vectors to produce a scalar. The formula is given by \( \mathbf{a} \cdot \mathbf{b} = \|\mathbf{a}\| \|\mathbf{b}\| \cos(\theta) \), where \( \theta \) is the angle between vectors \( \mathbf{a} \) and \( \mathbf{b} \). This mathematical operation has numerous applications, such as calculating the angle between vectors and finding projections.

Points to remember about the dot product:

• The result is always a scalar, a single numeric value representing magnitude, not direction.
• It’s commutative, meaning \( \mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a} \).
• If two vectors are perpendicular, their dot product is zero because \( \cos(90°) = 0 \).

Understanding dot products is essential for analyzing vector interactions, like those seen in physics and engineering problems.

Scalars and Vectors

In vector algebra, distinguishing between scalars and vectors is crucial. Scalars are quantities described by magnitude alone, such as temperature or distance. They are represented by real numbers. Vectors, on the other hand, have both magnitude and direction, like force or velocity, and are usually represented as arrows in space.

Here are key features:

• Scalars: Only magnitude. Examples include time, speed, and mass. You perform basic arithmetic operations on scalars like addition, subtraction, multiplication, and division.
• Vectors: Have direction and magnitude. Represented by notations such as \( \mathbf{v} \) or \( \langle x, y, z \rangle \). Operations include addition using the parallelogram law and scalar multiplication (scaling).

Recognizing the different roles of scalars and vectors is pivotal when performing vector operations. Misidentifying them can lead to math errors, such as incorrectly adding them together as seen in exercise b.

Vector Operations

Vector operations include a variety of calculations essential for handling vector quantities. Here are some crucial ones you'll encounter:

• Addition: Vectors are added by summing their respective components. If \( \mathbf{a} = \langle a_1, a_2 \rangle \) and \( \mathbf{b} = \langle b_1, b_2 \rangle \), then \( \mathbf{a} + \mathbf{b} = \langle a_1 + b_1, a_2 + b_2 \rangle \).
• Subtraction: Similar to addition, use component-wise subtraction.
• Scalar Multiplication: When a vector \( \mathbf{v} \) is multiplied by a scalar \( k \), each component of the vector is multiplied by \( k \), resulting in a vector scaled by \( k \).

It is essential to differentiate operations that result in a vector from those resulting in a scalar. In vector algebra, operations need to follow rules, such as dot products resulting in scalars, while vector addition or scalar multiplication results in a vector. Grasping these operations facilitates problem-solving across many fields, from physics to computer graphics.