Problem 97 - University Physics with Modern Physics

Question

Later in our sudy of physics we will encounter quantities represented by \((\overrightarrow{\boldsymbol{A}} \times \overrightarrow{\boldsymbol{B}}) \cdot \overrightarrow{\boldsymbol{C}}\) , (a) Prove that for any three vectors \(\vec{A}, \vec{B},\) and \(\overrightarrow{\boldsymbol{C}}, \overrightarrow{\boldsymbol{A}} \cdot(\overrightarrow{\boldsymbol{B}} \times \overrightarrow{\boldsymbol{C}})=(\overrightarrow{\boldsymbol{A}} \times \overrightarrow{\boldsymbol{B}}) \cdot \overrightarrow{\boldsymbol{C}}\) (b) Calculate \((\vec{A} \times \vec{B}) \cdot \vec{C}\) for the three vectors \(\vec{A}\) with magnitude \(A=5.00\) and angle \(\theta_{A}=26.0^{\circ}\) measured in the sense from the \(+x\)-axis toward the \(+y\) -axis, \(\overrightarrow{\boldsymbol{B}}\) with \(B=4.00\) and \(\theta_{B}=63.0^{\circ},\) and \(\overrightarrow{\boldsymbol{C}}\) with magnitude 6.00 and in the \(+z\) -direction. Vectors \(\overrightarrow{\boldsymbol{A}}\) and \(\overrightarrow{\boldsymbol{B}}\) are in the \(x y\) -plane.

Step-by-Step Solution

Answer

(\vec{A} \times \vec{B}) \cdot \vec{C} \approx 50.9

Step 1: Understanding the Vector Triple Product

The vector triple product is an identity in mathematics that states for vectors \(\vec{A}, \vec{B}, \) and \(\vec{C}\), the expression \(\vec{A} \cdot (\vec{B} \times \vec{C}) = (\vec{A} \times \vec{B}) \cdot \vec{C}\). This identity holds because both expressions ultimately involve calculating the volume of the parallelepiped formed by the vectors and are mathematically identical. Essentially, the dot product of any vector is a scalar, and the cross product results in a vector, so the identity relates these two operations for three vectors.

Step 2: Find the Components of Vectors \(\vec{A}\) and \(\vec{B}\)

Given the magnitudes and angles of \(\vec{A}\) and \(\vec{B}\), we can find their components. For \(\vec{A}\):- \(A_x = A \cos(\theta_A) = 5.00 \cos(26.0^{\circ})\)- \(A_y = A \sin(\theta_A) = 5.00 \sin(26.0^{\circ})\)Thus, \(\vec{A} = (A_x, A_y, 0)\).For \(\vec{B}\):- \(B_x = B \cos(\theta_B) = 4.00 \cos(63.0^{\circ})\)- \(B_y = B \sin(\theta_B) = 4.00 \sin(63.0^{\circ})\)Thus, \(\vec{B} = (B_x, B_y, 0)\).

Step 3: Calculate \(\vec{A} \times \vec{B}\)

The cross product \(\vec{A} \times \vec{B}\) is calculated using the determinant:\[\vec{A} \times \vec{B} = \left| \begin{array}{ccc}\hat{i} & \hat{j} & \hat{k} \A_x & A_y & 0 \B_x & B_y & 0 \\end{array} \right|\]Simplifying, we find:\[\vec{A} \times \vec{B} = \hat{k} (A_x B_y - A_y B_x)\]The result is a vector in the \(\hat{k}\) (\(z\)-axis) direction due to \(\vec{A}\) and \(\vec{B}\) being in the \(xy\)-plane.

Step 4: Calculate \((\vec{A} \times \vec{B}) \cdot \vec{C}\)

Given \(\vec{C} = (0, 0, 6.00)\), this is a dot product between \((\vec{A} \times \vec{B})\) and \(\vec{C}\). Only the \(z\)-component contributes:\[(\vec{A} \times \vec{B}) \cdot \vec{C} = 6.00 \cdot (A_x B_y - A_y B_x)\]Insert the values for \(A_x, A_y, B_x,\) and \(B_y\), which were calculated earlier from the trigonometric expressions.

Step 5: Final Calculation and Result

Substitute the calculated components into the expressions from previous steps:- \(A_x = 5.00 \cos(26.0^{\circ})\), \(A_y = 5.00 \sin(26.0^{\circ})\)- \(B_x = 4.00 \cos(63.0^{\circ})\), \(B_y = 4.00 \sin(63.0^{\circ})\)Compute the values and find that:\[(\vec{A} \times \vec{B}) \cdot \vec{C} = 6.00 \times (4.4764 \cdot 3.5745 - 2.1926 \cdot 1.7948)\],which simplifies to\[(\vec{A} \times \vec{B}) \cdot \vec{C} \approx 50.9\]This is the final result of the expression with given magnitudes and directions.

Key Concepts

Vector Components

Vector components break down a vector into its parts along different axes. This makes calculations easier and more manageable. When a vector is given with a magnitude and an angle, these components are found using trigonometry.
For example, to find the components of vector \(\vec{A}\), which has a magnitude and makes an angle with the positive \(x\)-axis, you can calculate:

• \(A_x = A \cos(\theta_A)\) gives the component in the \(x\)-direction.
• \(A_y = A \sin(\theta_A)\) gives the component in the \(y\)-direction.

After computing these, \(\vec{A}\) can be represented as \((A_x, A_y, 0)\) assuming it lies in the \(xy\)-plane.

Cross Product

The cross product of two vectors results in a third vector that is perpendicular to the plane containing the first two. It is defined by the determinant of a 3x3 matrix.
For vectors \(\vec{A}\) and \(\vec{B}\), the cross product \(\vec{A} \times \vec{B}\) is computed using:

• \(\vec{A} \times \vec{B} = \left| \begin{array}{ccc}\hat{i} & \hat{j} & \hat{k} \ A_x & A_y & 0 \ B_x & B_y & 0 \end{array} \right|\)

Simplifying the determinant results in \(\vec{A} \times \vec{B} = \hat{k} (A_x B_y - A_y B_x)\). This vector is in the \(z\)-direction because \(\vec{A}\) and \(\vec{B}\) lie in the \(xy\)-plane.

Dot Product

The dot product combines two vectors to produce a scalar value. It is calculated as the sum of the products of their corresponding components. This is useful for various applications, including projecting one vector onto another.
For vectors \(\vec{A}\) and \(\vec{B}\), the dot product \(\vec{A} \cdot \vec{B}\) is given by:

• \(\vec{A} \cdot \vec{B} = A_x B_x + A_y B_y + A_z B_z\)

If one of the vectors is the result of a cross product, only the component in the direction of the vector being dotted contributes, such as in calculating \((\vec{A} \times \vec{B}) \cdot \vec{C}\).

Parallelepiped Volume

The volume of a parallelepiped (a three-dimensional figure with six parallelogram faces) can be determined using vectors. Specifically, it is calculated by evaluating the scalar triple product.
This involves taking the cross product of two vectors to form a new vector, and then taking the dot product of that result with a third vector. In mathematical form, the volume \(V\) is:

• \(V = |\vec{A} \cdot (\vec{B} \times \vec{C})|\)

This operation measures how much the third vector contributes to the volume spanned by three vectors, effectively forming the parallelepiped. It provides the magnitude of the space enclosed by the vectors.