Problem 51 - University Physics with Modern Physics

Question

(a) Is the vector \((\hat{\imath}+\hat{j}+\hat{k})\) a unit vector? Justify your answer. (b) Can a unit vector have any components with magnitude greater than unity? Can it have any negative components? In each case justify your answer. (c) If \(\overrightarrow{\boldsymbol{A}}=a(3.0 \hat{\imath}+4.0 \hat{\mathbf{y}}),\) where \(\boldsymbol{a}\) is a constant, determine the value of \(a\) that makes \(\vec{A}\) a unit vector.

Step-by-Step Solution

Answer

(a) No, its magnitude is \(\sqrt{3}\). (b) It cannot have components > 1, but can be negative. (c) \(a = \frac{1}{5}\).

Step 1: Check if the vector is a unit vector

To determine if the vector \((\hat{\imath} + \hat{j} + \hat{k})\) is a unit vector, we need to calculate its magnitude. The magnitude of a vector \((a\hat{\imath} + b\hat{j} + c\hat{k})\) is computed as \(\sqrt{a^2 + b^2 + c^2}\). For our vector, \(a = 1, b = 1,\) and \(c = 1\). The magnitude is \(\sqrt{1^2 + 1^2 + 1^2} = \sqrt{3}\). Since \(\sqrt{3} eq 1\), the vector is not a unit vector.

Step 2: Components of a unit vector

A unit vector is defined as a vector with a magnitude of 1. Consequently, none of its components can have a magnitude greater than 1, as this would increase the overall magnitude beyond 1. However, a unit vector can have negative components. The direction of a component or vector does not impact its magnitude; it only changes its orientation.

Step 3: Find value of \(a\) for \(\overrightarrow{A}=a(3.0 \hat{\imath} + 4.0 \hat{\mathbf{y}})\) to be a unit vector

To find the value of \(a\) that makes \(\overrightarrow{A}\) a unit vector, we need the magnitude of \(\overrightarrow{A}\) to be 1. First calculate the magnitude: \(a\sqrt{(3.0)^2 + (4.0)^2} = a\sqrt{9 + 16} = a\sqrt{25} = 5a\). Set the magnitude equal to 1: \(5a = 1\). Solve for \(a\) by dividing both sides by 5: \(a = \frac{1}{5}\).

Step 4: Conclusion

Summarizing: (a) The vector \((\hat{\imath} + \hat{j} + \hat{k})\) is not a unit vector because its magnitude is \(\sqrt{3}\). (b) A unit vector cannot have components with magnitude greater than 1 but can have negative components. (c) The value of \(a\) required to make \(\overrightarrow{A}\) a unit vector is \(\frac{1}{5}\).

Key Concepts

Unit Vector

A unit vector is a vector that has a magnitude of exactly 1 unit. Its primary purpose is to denote direction without affecting the magnitude. Imagine it as an arrow that points in a direction but has a fixed length of 1. This concept is very important in physics and engineering to express directions efficiently.

To determine if a given vector is a unit vector, you calculate its magnitude and check if it equals 1. Magnitude is calculated using the formula: \\( \text{magnitude} = \sqrt{a^2 + b^2 + c^2} \). For instance, for the vector \((\hat{\imath} + \hat{j} + \hat{k})\), each component is 1, so the magnitude is \(\sqrt{1^2 + 1^2 + 1^2} = \sqrt{3}\). Because \(\sqrt{3} eq 1\), it is not a unit vector.

Unit vectors are useful because they allow you to separate direction from magnitude. When working with unit vectors, you can simply scale them to the desired magnitude for practical applications.

Magnitude of a Vector

The magnitude of a vector is its length, often thought of as the 'size' of the vector. For a vector, having components \(a\), \(b\), and \(c\), the magnitude is found using the formula: \[ \text{magnitude} = \sqrt{a^2 + b^2 + c^2} \]. This formula stems from the Pythagorean theorem applied in multiple dimensions.

The importance of vector magnitude lies in its various applications such as calculating speeds, forces, and distances in physics. Thus, understanding how to calculate the magnitude accurately is crucial in modeling real-world scenarios.

The exercise above involves confirming if a vector \((\hat{\imath} + \hat{j} + \hat{k})\) is a unit vector. We found its magnitude to be \(\sqrt{3}\), indicating that it is not a unit vector as its magnitude is not equal to 1. This step is key in determining whether a given vector, when scaled, can represent a true directionality without affecting size.

Vector Components

Every vector can be broken down into components that lie along the axes of a coordinate system. For example, a vector in three-dimensional space has three components along the x, y, and z axes. If a vector is expressed as \(a\hat{\imath} + b\hat{j} + c\hat{k}\), then \(a\), \(b\), and \(c\) are its components.

Understanding vector components allows us to analyze various attributes of the vector, such as direction and magnitude, separately. It’s crucial to remember that components can have negative values, which indicate direction. A negative component points in the opposite direction of the corresponding positive axis. However, for unit vectors, none of the components can exceed a magnitude of 1, or else the total magnitude would be greater than 1. Negative components are perfectly acceptable, as long as the overall magnitude remains 1.

Analyzing a vector based on its components helps solve problems related to direction and force distribution in physics. By altering components, you can represent any vector uniquely, making computations more convenient and interpretations more rigorous in practical scenarios.