Problem 6 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Suppose \(\overrightarrow{\mathrm{v}}\) is a vector of length 8 and the component of \(\overrightarrow{\mathbf{v}}\) in the direction of \(\overrightarrow{\mathrm{u}}\) is \(4 .\) (a) Can the angle between \(\overrightarrow{\mathrm{u}}\) and \(\overrightarrow{\mathrm{v}}\) be determined? If so, what is it? (b) Can the direction of \(\overrightarrow{\mathrm{u}}\) be determined? If so, what is it? (c) Can the length of \(\overrightarrow{\mathrm{u}}\) be determined? If so, what is it?

Step-by-Step Solution

Answer

(a) Yes, the angle can be determined. The angle between \(\overrightarrow{\mathrm{u}}\) and \(\overrightarrow{\mathrm{v}}\) is \(\frac{\pi}{3}\) or 60 degrees. (b) No, the direction of \(\overrightarrow{\mathrm{u}}\) cannot be determined from the given information. (c) No, the length of \(\overrightarrow{\mathrm{u}}\) cannot be determined from the given information.

Step 1: Determine the angle

To determine the angle between \(\overrightarrow{\mathrm{u}}\) and \(\overrightarrow{\mathrm{v}}\), we use the given magnitudes for \(\overrightarrow{\mathrm{v}}\) and component of \(\overrightarrow{\mathrm{v}}\) in the direction of \(\overrightarrow{\mathrm{u}}\). From the definition of the vector component, we know that 4 = 8*cos(θ). Solving this equation for the angle θ we get cos(θ) = 4 / 8 = 0.5. Inverse cosine function then yields θ = cos⁻¹(0.5) = \(\frac{\pi}{3}\) or 60 degrees.

Step 2: Determine the direction

The direction of a vector is not determined by its components in terms of other vectors. Therefore, the given information is not sufficient to determine the direction of \(\overrightarrow{\mathrm{u}}\). The direction depends on the specific orientation of \(\overrightarrow{\mathrm{u}}\) which has not been provided in this exercise.

Step 3: Determine the length

The length or magnitude of \(\overrightarrow{\mathrm{u}}\) cannot be determined from the given information. The component of \(\overrightarrow{\mathrm{v}}\) in the direction of \(\overrightarrow{\mathrm{u}}\) alone does not provide any information about the magnitude of \(\overrightarrow{\mathrm{u}}\). Therefore, the length of \(\overrightarrow{\mathrm{u}}\) cannot be determined from the given information.

Key Concepts

Vector Components

Vector components express a vector in terms of its influence in specific directions. Consider a vector \( \overrightarrow{\mathbf{v}} \). The component of \( \overrightarrow{\mathbf{v}} \) in another vector's direction, like \( \overrightarrow{\mathrm{u}} \), gives us the magnitude that \( \overrightarrow{\mathbf{v}} \) "projects" onto \( \overrightarrow{\mathrm{u}} \). This is computed through the dot product between the two vectors, which inherently involves the cosine of the angle between them.

Formally, if \( \overrightarrow{\mathbf{v}} \) has a length of 8 and the component in the direction of \( \overrightarrow{\mathrm{u}} \) is 4, we use the equation \( v_{\mathrm{u}} = \|\overrightarrow{\mathbf{v}}\| \cdot \cos(\theta) \). In our example, this yields \( 8 \cdot \cos(\theta) = 4 \).

This information reveals part of the interaction between the vectors, focusing on the angle \( \theta \), indicating their relative alignment.

Vector Magnitude

The magnitude of a vector, often visualized as its length, gives us a fundamental property of that vector. For a vector \( \overrightarrow{\mathbf{v}} \), its magnitude is typically noted as \( \|\overrightarrow{\mathbf{v}}\| \) and is always a non-negative number.

In the case of our exercise, \( \overrightarrow{\mathbf{v}} \) has a given magnitude of 8. This length is crucial when determining how it relates or projects onto another vector. When you multiply the magnitude by the cosine of the angle between the vectors, you achieve a scalar value, known as the vector component in a specific direction, which was given as 4 in our example.

Understanding vector magnitude helps in numerous applications, including physics, where it represents quantities like force or velocity.

Angle Between Vectors

The angle between two vectors is a key element in vector calculus, providing insight into their spatial relation. To find this angle, knowledge of the vectors' components and their interaction through the dot product is necessary.

In our problem, we used the equation derived from the vector component \( 8 \cdot \cos(\theta) = 4 \) to find \( \cos(\theta) = 0.5 \). Solving this using the inverse cosine function gives us \( \theta = \cos^{-1}(0.5) = \frac{\pi}{3} \) radians, or 60 degrees. This solution highlights how two vectors can be represented geometrically.

Angles between vectors are more than just mathematical curiosities; they have practical utilities in determining how two forces, velocities, or directions interact or oppose each other.

Vector Direction

Vector direction defines the orientation of a vector in space, pointing towards where its effect is headed. It is independent of its magnitude and often represented as an arrow in directional diagrams.

In our particular exercise, it was challenging to determine the direction of \( \overrightarrow{\mathrm{u}} \) since no specific orientation was given. Despite knowing the angle between \( \overrightarrow{\mathrm{v}} \) and \( \overrightarrow{\mathrm{u}} \), we lack sufficient data to pinpoint the exact direction, which typically requires either the coordinates or a reference angle.

Determining vector direction is essential in fields like navigation and physics, where getting the precise heading can be the difference between success and failure in problem-solving.