Problem 53 - Essential Calculus Early Transcendentals

Question

Suppose that \(\mathbf{a} \neq \mathbf{0}\). $$\begin{array}{l}{\text { (a) If } \mathbf{a} \cdot \mathbf{b}=\mathbf{a} \cdot \mathbf{c}, \text { does it follow that } \mathbf{b}=\mathbf{c} ?} \\\ {\text { (b) If } \mathbf{a} \times \mathbf{b}=\mathbf{a} \times \mathbf{c}, \text { does it follow that } \mathbf{b}=\mathbf{c} ?} \\ {\text { (c) If } \mathbf{a} \cdot \mathbf{b}=\mathbf{a} \cdot \mathbf{c} \text { and } \mathbf{a} \times \mathbf{b}=\mathbf{a} \times \mathbf{c}, \text { does it follow }} \\ {\text { that } \mathbf{b}=\mathbf{c} ?}\end{array}$$

Step-by-Step Solution

Answer

(a) No, (b) No, (c) Yes.

Step 1: Analyzing Case (a) - Dot Product Property

The dot product of two vectors \( \mathbf{a} \cdot \mathbf{b} \) gives a scalar result that is related to the angle between\( \mathbf{a} \) and \( \mathbf{b} \). If \( \mathbf{a} \cdot \mathbf{b} = \mathbf{a} \cdot \mathbf{c} \), it means the projections of \( \mathbf{b} \) and \( \mathbf{c} \) onto \( \mathbf{a} \) are the same. However, this does not necessarily mean \( \mathbf{b} = \mathbf{c} \), as there can be different vectors that project the same onto \( \mathbf{a} \), especially if \( \mathbf{b} - \mathbf{c} \) is perpendicular to \( \mathbf{a} \). Therefore, the answer is no.

Step 2: Analyzing Case (b) - Cross Product Property

The cross product \( \mathbf{a} \times \mathbf{b} \) gives a vector that is perpendicular to both \( \mathbf{a} \) and \( \mathbf{b} \). If \( \mathbf{a} \times \mathbf{b} = \mathbf{a} \times \mathbf{c} \), it implies that \( \mathbf{b} \) and \( \mathbf{c} \) are in the same plane passing through \( \mathbf{a} \). However, this doesn't ensure that \( \mathbf{b} = \mathbf{c} \), because \( \mathbf{b} - \mathbf{c} \) could be parallel to \( \mathbf{a} \). Therefore, the answer is no.

Step 3: Analyzing Case (c) - Combined Dot and Cross Product

Here we consider both conditions: \( \mathbf{a} \cdot \mathbf{b} = \mathbf{a} \cdot \mathbf{c} \) and \( \mathbf{a} \times \mathbf{b} = \mathbf{a} \times \mathbf{c} \). The first condition implies equal projections onto \( \mathbf{a} \), while the second implies that the perpendicular components (in the plane) are also equal. Since both conditions together ensure both parallel and perpendicular components relative to \( \mathbf{a} \) are equal, it follows that \( \mathbf{b} = \mathbf{c} \). Thus, the answer is yes.

Key Concepts

Dot Product

The dot product is a fundamental operation in vector algebra, providing valuable insights into the relationship between two vectors. When we calculate the dot product of vectors \( \mathbf{a} \) and \( \mathbf{b} \), we find the result \( \mathbf{a} \cdot \mathbf{b} = \| \mathbf{a} \| \| \mathbf{b} \| \cos \theta \), where \( \theta \) is the angle between the vectors. This result is a scalar.

• Projections: The dot product reveals how much of one vector “projects” onto another. It tells us about the component of one vector in the direction of the other.
• Identifying angles: When the dot product is zero, the vectors are perpendicular. Positive values mean vectors point generally in the same direction, while negative values indicate opposite directions.
• Equality condition: As observed in case (a), if \( \mathbf{a} \cdot \mathbf{b} = \mathbf{a} \cdot \mathbf{c} \), it merely states that the projections of \( \mathbf{b} \) and \( \mathbf{c} \) onto \( \mathbf{a} \) are the same. This condition alone does not guarantee that \( \mathbf{b} \) equals \( \mathbf{c} \).

Cross Product

The cross product is another critical operation in vector algebra that provides both directional and magnitude information about two vectors. For vectors \( \mathbf{a} \) and \( \mathbf{b} \), the cross product \( \mathbf{a} \times \mathbf{b} \) results in a vector. This vector is perpendicular to the plane formed by \( \mathbf{a} \) and \( \mathbf{b} \), with magnitude equal to \( \| \mathbf{a} \| \| \mathbf{b} \| \sin \theta \).

• Plane definition: This perpendicular vector defines the plane in which \( \mathbf{a} \) and \( \mathbf{b} \) reside.
• Zero result: If the cross product is zero, it indicates that vectors are either parallel or one of them is the zero vector.
• Identity condition: In case (b), if \( \mathbf{a} \times \mathbf{b} = \mathbf{a} \times \mathbf{c} \), it portrays that \( \mathbf{b} \) and \( \mathbf{c} \) align in the same plane with respect to \( \mathbf{a} \). However, on its own, this does not confirm \( \mathbf{b} = \mathbf{c} \).

Vector Projections

Vector projections involve representing a vector in terms of another vector's direction, fundamentally tied to the idea of components.

• Projection formula: The projection of vector \( \mathbf{b} \) onto vector \( \mathbf{a} \) is given by \( \text{proj}_\mathbf{a} \mathbf{b} = \frac{\mathbf{a} \cdot \mathbf{b}}{\| \mathbf{a} \|^2} \mathbf{a} \). This shows how much \( \mathbf{b} \) “lies along” \( \mathbf{a} \).
• Combining information: In situation (c), both the dot and cross product equations hold true. Thus, \( \mathbf{b} \) and \( \mathbf{c} \) not only share the same projections (parallel components) onto \( \mathbf{a} \), but also their perpendicular components (cross product) are equal, ultimately leading to \( \mathbf{b} = \mathbf{c} \).
• Useful in applications: Vector projections are particularly useful in physics for decomposing forces and in computer graphics for rendering light and shadows.