Problem 36 - University Physics with Modern Physics

Question

In each case, find the \(x\)- and \(y\)-components of vector \(\overrightarrow{A}\): (a) \(\overrightarrow{A}\) = 5.0\(\hat{\imath}\) \(-\) 6.3\(\hat{\jmath}\); (b) \(\overrightarrow{A}\) = 11.2\(\hat{\jmath}\) \(-\) 9.91\(\hat{\imath}\); (c) \(\overrightarrow{A}\) = \(-\)15.0\(\hat{\imath}\) \(+\) 22.4\(\hat{\jmath}\) ; (d) \(\overrightarrow{A}\) = 5.0\(\overrightarrow{B}\), where \(\overrightarrow{B}\) = 4\(\hat{\imath}\) \(+\) 6\(\hat{\jmath}\).

Step-by-Step Solution

Answer

(a) x: 5.0, y: -6.3; (b) x: -9.91, y: 11.2; (c) x: -15.0, y: 22.4; (d) x: 20.0, y: 30.0.

Step 1: Extract x and y components for vector (a)

For vector \( \overrightarrow{A} = 5.0\hat{\imath} - 6.3\hat{\jmath} \), the \( x \)-component is the coefficient of \( \hat{\imath} \), which is 5.0, and the \( y \)-component is the coefficient of \( \hat{\jmath} \), which is -6.3.

Step 2: Extract x and y components for vector (b)

For vector \( \overrightarrow{A} = 11.2\hat{\jmath} - 9.91\hat{\imath} \), swap the order to standard form: \( \overrightarrow{A} = -9.91\hat{\imath} + 11.2\hat{\jmath} \). The \( x \)-component is -9.91 and the \( y \)-component is 11.2.

Step 3: Extract x and y components for vector (c)

For vector \( \overrightarrow{A} = -15.0\hat{\imath} + 22.4\hat{\jmath} \), the \( x \)-component is -15.0 and the \( y \)-component is 22.4.

Step 4: Calculate components for vector (d)

Given \( \overrightarrow{A} = 5.0\overrightarrow{B} \) and \( \overrightarrow{B} = 4\hat{\imath} + 6\hat{\jmath} \), multiply each component of \( \overrightarrow{B} \) by 5.0. The \( x \)-component becomes \( 5.0 \times 4 = 20.0 \), and the \( y \)-component becomes \( 5.0 \times 6 = 30.0 \).

Key Concepts

Physics Vectors

In physics, vectors are essential tools used to represent quantities that possess both magnitude and direction. Imagine vectors as arrows: the length of the arrow represents the magnitude, while the direction in which the arrow points signifies its direction. In mathematical terms, these vectors can be broken down into components, such as the x-component and y-component in a two-dimensional space.

• Magnitude: Size or length of the vector.
• Direction: The path that the vector follows.
• Components: Parts of the vector along the axes of the coordinate system.

When addressing physics problems, understanding vectors assists us in decomposing complex movements into simpler parts. This often makes calculations much more accessible and intuitive.

Vector Addition

Vector addition is straightforward when you comprehend vectors as displacements on a coordinate plane. To add vectors, one should add their respective components. Begin by arranging the vectors such that their tails meet consecutively, maintaining an endpoint-to-beginning alignment for accuracy.

• Add the x-components together to find the new x-component.
• Add the y-components to get the resulting y-component.
• The sum will yield a new vector, the result of the addition.

This method is often visualized through the ‘tip-to-tail’ method. But, when we need computational precision, component-wise addition is the go-to. Thus, vector addition becomes quite intuitive and visual.

Coordinate System

The coordinate system serves as a backdrop for defining positions in space through numerical coordinates. In a two-dimensional plane, these positions are often specified using x and y coordinates, which form the axes.

• The x-axis typically runs horizontally.
• The y-axis usually runs vertically.
• Together, these axes intersect at a point known as the origin.

Using such a system, vectors can be conveniently represented and manipulated. Vectors' components align neatly along these axes, allowing for the separation of problems into manageable one-dimensional tasks for easier calculation.

Vector Multiplication

Vector multiplication involves scaling a vector by a numerical factor, which affects the vector's magnitude while leaving its direction unchanged, assuming the factor is positive. This type of multiplication is crucial when elements need to match real-world measurements or remain proportional.

• Scaling: When multiplying a vector by a scalar (a number), both its x and y components are multiplied by that number.
• Magnitude Change: Only the vector's magnitude changes; positive scaling factors keep the direction the same, whereas negative factors reverse it.

For instance, if we have a vector \( \overrightarrow{B} \) represented as \( 4\hat{\imath} + 6\hat{\jmath} \), multiplying by 5 results in a new vector with components \( 20\hat{\imath} + 30\hat{\jmath} \). Such manipulations are effortless when one understands the fundamentals of vector multiplication.