Problem 24 - College Physics

Question

(a) If each of the two components \((x\) and \(y\) ) of a vector are doubled, (1) the vector's magnitude doubles, but the direction remains unchanged; (2) the vector's magnitude remains unchanged, but the direction angle doubles; or (3) both the vector's magnitude and direction angle double. (b) If the \(x\) - and \(y\) -components of a vector of \(10 \mathrm{~m}\) at \(45^{\circ}\) are tripled, what is the new vector?

Step-by-Step Solution

Answer

(a) The vector's magnitude doubles; direction remains unchanged. (b) New vector is 30 m at 45°.

Step 1: Understanding the Problem

We are given a vector where both components (\(x\) and \(y\)) can be altered. We need to determine what happens to the vector's magnitude and direction when the components are doubled or tripled.

Step 2: Defining Vector Magnitude and Direction

The magnitude of a vector \( \vec{v} = (x, y) \) is calculated using the formula: \( ||\vec{v}|| = \sqrt{x^2 + y^2} \). The direction, or angle \( \theta \), is given by \( \theta = \tan^{-1}(\frac{y}{x}) \).

Step 3: Component Doubling (Part a)

If each component, \(x\) and \(y\), of a vector is doubled, the new vector becomes \( (2x, 2y) \). The magnitude \( ||\vec{v}|| = \sqrt{(2x)^2 + (2y)^2} = 2 \sqrt{x^2 + y^2} \), indicating the magnitude doubles. The direction remains \( \theta = \tan^{-1}(\frac{2y}{2x}) = \tan^{-1}(\frac{y}{x}) \), thus it is unchanged.

Step 4: Conclusion for Part (a)

Since the magnitude of the vector doubles and the direction is unchanged, the correct option is (1): the vector's magnitude doubles, but the direction remains unchanged.

Step 5: Component Tripling for New Vector (Part b)

Given a vector of 10 m at 45 degrees, its components are \( x = 10 \cos(45^{\circ}) = 10 \times \frac{\sqrt{2}}{2} = 5\sqrt{2} \) and \( y = 10 \sin(45^{\circ}) = 10 \times \frac{\sqrt{2}}{2} = 5\sqrt{2} \). If the components are tripled, the new components are \( 15\sqrt{2} \) each.

Step 6: Calculate New Vector Magnitude and Direction for Part (b)

The new vector becomes \((15\sqrt{2}, 15\sqrt{2})\). The magnitude is \( \sqrt{(15\sqrt{2})^2 + (15\sqrt{2})^2} = \sqrt{450 + 450} = 30\). The direction stays the same because \( \theta = \tan^{-1}(\frac{15\sqrt{2}}{15\sqrt{2}}) = \tan^{-1}(1) = 45^{\circ} \).

Key Concepts

Vector Magnitude

The magnitude of a vector is essentially a measure of its "length" or size in a geometric space. When we think about vectors, we usually visualize them as arrows pointing a certain direction in a plane or three-dimensional space. The length of this arrow is what we call the magnitude.
To calculate the magnitude of a two-dimensional vector, let's say \( \vec{v} = (x, y) \), we use the Pythagorean Theorem. The formula is:

• \( ||\vec{v}|| = \sqrt{x^2 + y^2} \)

This formula essentially takes the square root of the sum of squares of its components, \(x\) and \(y\).
For instance, if we have a vector with x-component 3 and y-component 4, the magnitude would be \( \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5 \). Just like that, we've calculated the size of our vector.

Vector Components

Vector components are the projections of a vector along the axes of the coordinate system. Think of a vector as having two independent parts: one in the x-direction and one in the y-direction. These parts or projections are what we call the vector's components. For a vector \( \vec{v} = (x, y) \), the components describe how far the vector stretches in each direction.

• The x-component might be represented as \( ||\vec{v}|| \cos(\theta) \)
• The y-component as \( ||\vec{v}|| \sin(\theta) \)

An interesting fact: if you alter these components by a factor, say doubling them, you essentially change the vector's representation in space. Doubling components \((2x, 2y)\) results in doubling the vector's magnitude without impacting its direction. So vector components are crucial for understanding how alterations affect a vector's properties.

Vector Direction

The direction of a vector specifies the way it is pointing, often represented as an angle with respect to a horizontal line or axis. Imagine standing at an origin point and looking in the direction the vector points. This angle is typically measured in degrees or radians from the positive x-axis. The angle, \( \theta \), can be determined using trigonometric functions among the vector's components:

• \( \theta = \tan^{-1}(\frac{y}{x}) \)

Even if the magnitude changes, as long as the proportion between the x-component and y-component remains the same, the direction stays unchanged. This means, for vectors like \( (x, y) \) and \( (2x, 2y) \), both point in the same direction. Thus, understanding vector direction is key in physics and engineering to predict vector behavior under various transformations.

Trigonometric Functions

Trigonometric functions are mathematical tools that relate the angles and sides of right-angled triangles. They are crucial in vector mathematics to determine vector directions and components. The primary trigonometric functions used in vectors are sine, cosine, and tangent.
Each function has a specific role:

• \( \sin(\theta) \) relates the opposite side to the hypotenuse
• \( \cos(\theta) \) connects the adjacent side to the hypotenuse
• \( \tan(\theta) \) combines the opposite side over the adjacent side

For vectors, these functions help us break down a vector into its x and y components. This is done via the relationships:
\[ x = ||\vec{v}|| \cos(\theta) \quad \text{and} \quad y = ||\vec{v}|| \sin(\theta) \]
Understanding these functions aids in visualizing how vectors operate in different coordinate systems, providing a basis for complex analyses and problem-solving in mathematics and physics.