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Chapter 8: Problem 107

Using two mathematical representations of the cross product, find the angle between the two vectors \(\vec{A}\) and \(\vec{B}\) : $$ \begin{array}{r} \vec{A}=3 \hat{x}+4 \hat{y}+2 \hat{z} \\ \vec{B}=5 \hat{x}-2 \hat{y}-3 \hat{z} \end{array} $$

Short Answer

Expert verified
Checking the solution steps, the angle \(\Theta\) between the vectors \(\vec{A}\) and \(\vec{B}\) is found by using the cross product's magnitude and applying it to the formula to calculate the angle.

Step by step solution

01

Calculate the Cross Product

To calculate the cross product, write out the i, j and k components as the first row of a 3x3 matrix, put the components of \(\vec{A}\) as second row and the components of \(\vec{B}\) as the third row. Then calculate the determinant of this matrix, which gives the components for the cross product of \(\vec{A}\) and \(\vec{B}\).
02

Calculate the magnitudes of \(\vec{A}\), \(\vec{B}\) and \(\vec{A}\times\vec{B}\)

Calculate the magnitudes (or lengths) of \(\vec{A}\), \(\vec{B}\), and \(\vec{A}\times\vec{B}\). The magnitude of a vector can be calculated by taking the square root of the sum of the squares of its components.
03

Calculate the Angle

Substitute the magnitudes of \(\vec{A}\), \(\vec{B}\) and \(\vec{A}\times\vec{B}\) into the formula for the angle between the two vectors given that \[\sin\Theta = \frac{{||\vec{A} \times \vec{B}||}}{{||\vec{A}|| \cdot ||\vec{B}||}}\] then use the inverse sine function to find the angle \(\Theta\).

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Key Concepts

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

Vector Magnitude
To find the magnitude of a vector, you sum the squares of its components and then take the square root of this sum. This process gives us a measure of a vector's length, which is crucial for understanding its size in space.
For example, consider the vector \(\vec{A} = 3 \hat{x} + 4 \hat{y} + 2 \hat{z}\).
To find the magnitude of \(\vec{A}\), calculate as follows:
  • Square each component: \(3^2 = 9\), \(4^2 = 16\), \(2^2 = 4\)
  • Sum the squares: \(9 + 16 + 4 = 29\)
  • Take the square root: \(\sqrt{29}\approx 5.39\)
The magnitude is approximately 5.39 units.
This method applies to any vector, providing a standard way to gauge its length.
Determinant Calculation
A determinant provides a scalar value that is very helpful in calculating the cross product of two vectors. For vectors \(\vec{A}\) and \(\vec{B}\), arranging them in a 3x3 matrix with unit vectors \(\hat{i}, \hat{j}, \hat{k}\) in the first row helps in this process.
The matrix setup looks like this:
  • First row: \(\hat{i}, \hat{j}, \hat{k}\)
  • Second row: components of \(\vec{A}\): \(3, 4, 2\)
  • Third row: components of \(\vec{B}\): \(5, -2, -3\)
To find the cross product \(\vec{A} \times \vec{B}\), calculate the determinant of this matrix such as:
  • Calculate \(\hat{i}(4(-3) - (-2)2) - \hat{j}(3(-3) - 2(5)) + \hat{k}(3(-2) - 4(5))\)
  • Simplify to get components of the cross product vector
The determinant gives a step-by-step solution to finding the orientation and scaling of the vector product.
Angle Between Vectors
The angle between two vectors can be found using the cross product and magnitudes of the vectors. This relationship is represented by the formula for the sine of the angle:
\[\sin \Theta = \frac{{||\vec{A} \times \vec{B}||}}{{||\vec{A}|| \cdot ||\vec{B}||}}\]
Here,
  • \(||\vec{A} \times \vec{B}||\) is the magnitude of the cross product
  • \(||\vec{A}||\) and \(||\vec{B}||\) are magnitudes of \(\vec{A}\) and \(\vec{B}\)
Once you have these values, substitute them into the formula.
Then, use the inverse sine function (\(\sin^{-1}\) or arcsin) to find the angle \(\Theta\).
Using this method gives you not only the size but also an understanding of the spatial relationship between two vectors.

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Most popular questions from this chapter

A typical adult can deliver about \(10 \mathrm{~N} \cdot \mathrm{m}\) of torque when attempting to open a twist-off cap on a bottle. What is the maximum force that the average person can exert with his fingers if most bottle caps are about \(2 \mathrm{~cm}\) in diameter?

\(\bullet\) Bob and Lily are riding on a merry-go-round. Bob rides on a horse at toward the outer edge of a circular platform and Lily rides on a horse toward the center of the circular platform. When the merry-go-round is rotating at a constant angular speed \(\omega\), Bob's speed \(v\) is A. exactly half as much as Lily's. B. larger than Lily's. C. smaller than Lily's. D. the same as Lily's. E. exactly twice as much as Lily's.

Describe any inconsistencies in the following statement: The units of torque are \(\mathrm{N} \cdot \mathrm{m}\), but that's not the same as the units of energy.

Analyze the following statement and determine if there are any physical inconsistencies: While rotating a ball on the end of a string of length \(L\), the rotational kinetic energy remains constant as long as the length and angular speed are fixed. When the ball is pulled inward and the length of the string is shortened, the rotational kinetic energy will remain constant due to conservation of energy, but the angular momentum will not, because there is an external force acting on the ball to pull it inward. The moment of inertia and angular speed will, of course, remain the same throughout the process because the ball is rotating in the same plane throughout the motion.

Describe how many unique ways a vector cross product \((\vec{C}=\vec{A} \times \vec{B})\) can equal zero.
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