Problem 36 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find a simple expression for each of the following for an arbitrary vector \(\mathbf{u}\). (a) \(\operatorname{proj}_{\mathbf{u}}(-\mathbf{u})\) (b) \(\operatorname{proj}_{-\mathbf{u}}(-\mathbf{u})\)

Step-by-Step Solution

Answer

Both projections result in \( -\mathbf{u} \).

Step 1: Understand the Projection Formula

The projection of a vector \( \mathbf{a} \) onto another vector \( \mathbf{b} \) is given by:\[\operatorname{proj}_{\mathbf{b}}\mathbf{a} = \frac{\mathbf{a} \cdot \mathbf{b}}{\mathbf{b} \cdot \mathbf{b}} \mathbf{b}.\]This formula projects \( \mathbf{a} \) along \( \mathbf{b} \). "\( \cdot \)" denotes the dot product.

Step 2: Apply the Formula to (a)

For \( \operatorname{proj}_{\mathbf{u}}(-\mathbf{u}) \):1. Identify \( \mathbf{a} = -\mathbf{u} \) and \( \mathbf{b} = \mathbf{u} \).2. Calculate the dot product: \((-\mathbf{u}) \cdot \mathbf{u} = - \mathbf{u} \cdot \mathbf{u} = -\| \mathbf{u} \|^2\).3. The projection is:\[\operatorname{proj}_{\mathbf{u}}(-\mathbf{u}) = \frac{-\| \mathbf{u} \|^2}{\| \mathbf{u} \|^2} \mathbf{u} = -\mathbf{u}.\]

Step 3: Apply the Formula to (b)

For \( \operatorname{proj}_{-\mathbf{u}}(-\mathbf{u}) \):1. Identify \( \mathbf{a} = -\mathbf{u} \) and \( \mathbf{b} = -\mathbf{u} \).2. Calculate the dot product: \((-\mathbf{u}) \cdot (-\mathbf{u}) = \mathbf{u} \cdot \mathbf{u} = \| \mathbf{u} \|^2\).3. The projection is:\[\operatorname{proj}_{-\mathbf{u}}(-\mathbf{u}) = \frac{\| \mathbf{u} \|^2}{\| \mathbf{u} \|^2} (-\mathbf{u}) = -\mathbf{u}.\]

Key Concepts

Dot Product

The dot product is a foundational operation in vector mathematics. It is the sum of the products of the corresponding components of two vectors. If you have vectors \( \mathbf{a} = (a_1, a_2, a_3) \) and \( \mathbf{b} = (b_1, b_2, b_3) \), the dot product is calculated as:

• \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 \).

The dot product is also related to the cosine of the angle \( \theta \) between the vectors. The formula is:

• \( \mathbf{a} \cdot \mathbf{b} = \| \mathbf{a} \| \| \mathbf{b} \| \cos(\theta) \).

This property makes the dot product useful for determining angles between vectors and identifying orthogonal vectors, where the dot product is zero. It's often used in projection formulas and various computational techniques in physics and engineering.
Understanding how to calculate the dot product is crucial before moving on to more complex vector operations.

Projection Formula

The projection formula helps find the component of one vector along the direction of another vector. The projection of vector \( \mathbf{a} \) onto vector \( \mathbf{b} \) is given by:

• \( \operatorname{proj}_{\mathbf{b}}\mathbf{a} = \frac{\mathbf{a} \cdot \mathbf{b}}{\mathbf{b} \cdot \mathbf{b}} \mathbf{b} \).

This formula results in a vector that lies along \( \mathbf{b} \) and represents how much of \( \mathbf{a} \) stretches in the direction of \( \mathbf{b} \). If \( \mathbf{a} \) and \( \mathbf{b} \) are parallel, the projection will be equal to \( \mathbf{a} \) itself scaled by some factor.
The formula uses the dot product in both the numerator and denominator to obtain a scalar that scales \( \mathbf{b} \). Therefore, understanding dot products is essential for mastering projections.
A special case to note is when the projection of a vector onto itself is considered. Here, the projection vector is equal to the original vector due to the scaling factor becoming one.

Vector Operations

Vector operations include various mathematical manipulations such as addition, subtraction, and scalar multiplication. These operations obey specific algebraic rules making them predictable and reliable for solving geometric and scientific problems.

• Addition: For vectors \( \mathbf{a} = (a_1, a_2, a_3) \) and \( \mathbf{b} = (b_1, b_2, b_3) \), the sum is \( \mathbf{a} + \mathbf{b} = (a_1 + b_1, a_2 + b_2, a_3 + b_3) \).
• Subtraction: Given vectors \( \mathbf{a} \) and \( \mathbf{b} \), the difference is \( \mathbf{a} - \mathbf{b} = (a_1 - b_1, a_2 - b_2, a_3 - b_3) \).
• Scalar Multiplication: If \( c \) is a scalar and \( \mathbf{a} = (a_1, a_2, a_3) \), the product is \( c\mathbf{a} = (ca_1, ca_2, ca_3) \).

Each of these operations allows for the combination or modification of vectors to examine directional and magnitude aspects of vector quantities in physical contexts. Mastery of these operations is critical in fields like physics, engineering, and computer graphics. The elegance of vector operations lies in their simplicity and adaptability to various dimensional spaces.