Problem 93 - Precalculus with Limits

Question

Think About It What can be said about the vectors \(\mathbf{u}\) and \(\mathbf{v}\) under each condition? (a) The projection of \(\mathbf{u}\) onto \(\mathbf{v}\) equals \(\mathbf{u}\) . (b) The projection of \(\mathbf{u}\) onto \(\mathbf{v}\) equals \(\mathbf{0}\) .

Step-by-Step Solution

Answer

In condition (a), vectors \(\mathbf{u}\) and \(\mathbf{v}\) are in the same direction while in condition (b), vectors \(\mathbf{u}\) and \(\mathbf{v}\) are perpendicular to each other.

Step 1: Interpret Condition (a)

In condition (a), the projection of \(\mathbf{u}\) onto \(\mathbf{v}\) equals \(\mathbf{u}\). This means that vector \(\mathbf{u}\) lies entirely on the line formed by vector \(\mathbf{v}\). This is possible only if vectors \(\mathbf{u}\) and \(\mathbf{v}\) are in the same direction.

Step 2: Interpret Condition (b)

In condition (b), the projection of \(\mathbf{u}\) onto \(\mathbf{v}\) equals \(\mathbf{0}\). This implies that vector \(\mathbf{u}\) is perpendicular to vector \(\mathbf{v}\) because only in this case, the projection of a non-zero vector can be zero.

Key Concepts

Vectors

Vectors are fundamental objects in mathematics and physics. They have both magnitude (length) and direction. A vector is often represented as a directed line segment, where the length represents the magnitude and the arrow indicates the direction.
In mathematical notation, vectors are usually denoted by boldface letters, like \( \mathbf{u} \) and \( \mathbf{v} \).

• Magnitude: The size or length of the vector. It is often calculated using the Euclidean distance formula.
• Direction: The orientation of the vector in space, which can be described using angles or coordinate points.

Understanding vectors goes beyond their representation. Their role in various fields, like physics, engineering, and computer graphics, showcases their use in translating and scaling mathematical ideas into practical applications. They are used to represent quantities that have both a magnitude and direction, such as forces and velocities.

Orthogonality

Orthogonality is a key concept in understanding vector projections. Two vectors are orthogonal, or perpendicular, when the angle between them is 90 degrees. This results in their dot product being zero.
In simpler terms, if two vectors \( \mathbf{u} \) and \( \mathbf{v} \) are orthogonal, they do not influence each other. This lack of influence is reflected in vector projections:

• If the projection of \( \mathbf{u} \) onto \( \mathbf{v} \) is the zero vector \( \mathbf{0} \), then \( \mathbf{u} \) is orthogonal to \( \mathbf{v} \).

This orthogonality criterion helps in determining when projections yield no component along the direction of the vector being projected onto. In many analytical tasks, recognizing orthogonal vectors aids in simplifying complex vector algebra and geometry problems.

Directional Vectors

Directional vectors play a crucial role when analyzing vector projections. They determine the line along which a vector projection falls. In the context of projections, the directional vector \( \mathbf{v} \) indicates the direction onto which another vector \( \mathbf{u} \) is projected.

• When the projection of \( \mathbf{u} \) onto \( \mathbf{v} \) is \( \mathbf{u} \) itself, \( \mathbf{u} \) is in line with \( \mathbf{v} \). This indicates they are directional twins.
• If this projection is zero, \( \mathbf{u} \) holds no component in the direction of \( \mathbf{v} \), making them orthogonal as discussed.

Directional vectors serve as the backbone in determining how vectors align or mismatch within a vector space, shaping the basis for more complex vector-related tasks and operations.