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Chapter 1: Problem 36

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\(\hat{\jmath}\).

Short Answer

Expert verified
(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 by step solution

01

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.
02

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.
03

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.
04

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 \( 5.0 \times 6 = 30.0 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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 鈥榯ip-to-tail鈥� 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.

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Most popular questions from this chapter

As you eat your way through a bag of chocolate chip cookies, you observe that each cookie is a circular disk with a diameter of 8.50 \(\pm\) 0.02 cm and a thickness of 0.050 \(\pm\) 0.005 cm. (a) Find the average volume of a cookie and the uncertainty in the volume. (b) Find the ratio of the diameter to the thickness and the uncertainty in this ratio.

A disoriented physics professor drives 3.25 km north, then 2.20 km west, and then 1.50 km south. Find the magnitude and direction of the resultant displacement, using the method of components. In a vector-addition diagram (roughly to scale), show that the resultant displacement found from your diagram is in qualitative agreement with the result you obtained by using the method of components.

While driving in an exotic foreign land, you see a speed limit sign that reads 180,000 furlongs per fortnight. How many miles per hour is this? (One furlong is \\(\frac{1}{8}\\) mile, and a fortnight is 14 days. A furlong originally referred to the length of a plowed furrow.)

According to the label on a bottle of salad dressing, the volume of the contents is 0.473 liter (L). Using only the conversions 1 (L) = 1000 cm\(^3\) and 1 in. = 2.54 cm, express this volume in cubic inches.

Let \(\theta\) be the angle that the vector \(\overrightarrow{A}\) makes with the \(+$$x\)-axis, measured counterclockwise from that axis. Find angle \(\theta\) for a vector that has these components: (a) A\(_x\) = 2.00 m, A\(_y\) = \(-\)1.00 m; (b) A\(_x\) = 2.00 m, A\(_y\) = 1.00 m; (c) A\(_x\) = \(-\)2.00 m, A\(_y\) = 1.00 m; (d) A\(_x\) = -2.00 m, A\(_y\) = -1.00 m.
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