Answer :
Let's solve the problem step-by-step:
We have three vectors:
1. [tex]\(\vec{A} = 2.00 \hat{i} + 2.00 \hat{j} - 3.00 \hat{k}\)[/tex]
2. [tex]\(\vec{B} = -3.00 \hat{i} + 4.00 \hat{j} + 4.00 \hat{k}\)[/tex]
3. [tex]\(\vec{C} = 7.00 \hat{i} - 8.00 \hat{j}\)[/tex]
We need to compute [tex]\(2 \cdot \vec{C} \cdot (2 \vec{A} \times \vec{B})\)[/tex].
### Step 1: Calculate [tex]\(2 \vec{A}\)[/tex]
Multiply each component of [tex]\(\vec{A}\)[/tex] by 2:
[tex]\[
2 \vec{A} = 2 \cdot (2.00 \hat{i} + 2.00 \hat{j} - 3.00 \hat{k}) = 4.00 \hat{i} + 4.00 \hat{j} - 6.00 \hat{k}
\][/tex]
### Step 2: Calculate the cross product [tex]\(2 \vec{A} \times \vec{B}\)[/tex]
The cross product of two vectors [tex]\(\vec{u} = u_1 \hat{i} + u_2 \hat{j} + u_3 \hat{k}\)[/tex] and [tex]\(\vec{v} = v_1 \hat{i} + v_2 \hat{j} + v_3 \hat{k}\)[/tex] is determined by the determinant of the following matrix:
[tex]\[
\vec{u} \times \vec{v} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \end{vmatrix}
\][/tex]
For [tex]\(2 \vec{A} = 4.00 \hat{i} + 4.00 \hat{j} - 6.00 \hat{k}\)[/tex] and [tex]\(\vec{B} = -3.00 \hat{i} + 4.00 \hat{j} + 4.00 \hat{k}\)[/tex], we have:
[tex]\[
2 \vec{A} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 4.00 & 4.00 & -6.00 \\ -3.00 & 4.00 & 4.00 \end{vmatrix}
\][/tex]
Calculating the determinant:
[tex]\[
= \hat{i}(4.00 \cdot 4.00 - 4.00 \cdot (-6.00)) - \hat{j}(4.00 \cdot 4.00 - (-6.00) \cdot (-3.00)) + \hat{k}(4.00 \cdot 4.00 - 4.00 \cdot (-3.00))
\][/tex]
[tex]\[
= \hat{i}(16.00 + 24.00) - \hat{j}(16.00 - 18.00) + \hat{k}(16.00 + 12.00)
\][/tex]
[tex]\[
= 40.00 \hat{i} + 2.00 \hat{j} + 28.00 \hat{k}
\][/tex]
### Step 3: Calculate [tex]\(\vec{C} \cdot (2 \vec{A} \times \vec{B})\)[/tex]
For [tex]\(\vec{C} = 7.00 \hat{i} - 8.00 \hat{j} + 0.00 \hat{k}\)[/tex], take the dot product with the cross product:
[tex]\[
\vec{C} \cdot (2 \vec{A} \times \vec{B}) = (7.00 \times 40.00) + (-8.00 \times 2.00) + (0.00 \times 28.00)
\][/tex]
[tex]\[
= 280.00 - 16.00 + 0.00 = 264.00
\][/tex]
### Step 4: Calculate [tex]\(2 \cdot \vec{C} \cdot (2 \vec{A} \times \vec{B})\)[/tex]
Multiply the dot product by 2:
[tex]\[
2 \cdot 264.00 = 528.00
\][/tex]
So, the final result is [tex]\(528.00\)[/tex].
We have three vectors:
1. [tex]\(\vec{A} = 2.00 \hat{i} + 2.00 \hat{j} - 3.00 \hat{k}\)[/tex]
2. [tex]\(\vec{B} = -3.00 \hat{i} + 4.00 \hat{j} + 4.00 \hat{k}\)[/tex]
3. [tex]\(\vec{C} = 7.00 \hat{i} - 8.00 \hat{j}\)[/tex]
We need to compute [tex]\(2 \cdot \vec{C} \cdot (2 \vec{A} \times \vec{B})\)[/tex].
### Step 1: Calculate [tex]\(2 \vec{A}\)[/tex]
Multiply each component of [tex]\(\vec{A}\)[/tex] by 2:
[tex]\[
2 \vec{A} = 2 \cdot (2.00 \hat{i} + 2.00 \hat{j} - 3.00 \hat{k}) = 4.00 \hat{i} + 4.00 \hat{j} - 6.00 \hat{k}
\][/tex]
### Step 2: Calculate the cross product [tex]\(2 \vec{A} \times \vec{B}\)[/tex]
The cross product of two vectors [tex]\(\vec{u} = u_1 \hat{i} + u_2 \hat{j} + u_3 \hat{k}\)[/tex] and [tex]\(\vec{v} = v_1 \hat{i} + v_2 \hat{j} + v_3 \hat{k}\)[/tex] is determined by the determinant of the following matrix:
[tex]\[
\vec{u} \times \vec{v} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \end{vmatrix}
\][/tex]
For [tex]\(2 \vec{A} = 4.00 \hat{i} + 4.00 \hat{j} - 6.00 \hat{k}\)[/tex] and [tex]\(\vec{B} = -3.00 \hat{i} + 4.00 \hat{j} + 4.00 \hat{k}\)[/tex], we have:
[tex]\[
2 \vec{A} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 4.00 & 4.00 & -6.00 \\ -3.00 & 4.00 & 4.00 \end{vmatrix}
\][/tex]
Calculating the determinant:
[tex]\[
= \hat{i}(4.00 \cdot 4.00 - 4.00 \cdot (-6.00)) - \hat{j}(4.00 \cdot 4.00 - (-6.00) \cdot (-3.00)) + \hat{k}(4.00 \cdot 4.00 - 4.00 \cdot (-3.00))
\][/tex]
[tex]\[
= \hat{i}(16.00 + 24.00) - \hat{j}(16.00 - 18.00) + \hat{k}(16.00 + 12.00)
\][/tex]
[tex]\[
= 40.00 \hat{i} + 2.00 \hat{j} + 28.00 \hat{k}
\][/tex]
### Step 3: Calculate [tex]\(\vec{C} \cdot (2 \vec{A} \times \vec{B})\)[/tex]
For [tex]\(\vec{C} = 7.00 \hat{i} - 8.00 \hat{j} + 0.00 \hat{k}\)[/tex], take the dot product with the cross product:
[tex]\[
\vec{C} \cdot (2 \vec{A} \times \vec{B}) = (7.00 \times 40.00) + (-8.00 \times 2.00) + (0.00 \times 28.00)
\][/tex]
[tex]\[
= 280.00 - 16.00 + 0.00 = 264.00
\][/tex]
### Step 4: Calculate [tex]\(2 \cdot \vec{C} \cdot (2 \vec{A} \times \vec{B})\)[/tex]
Multiply the dot product by 2:
[tex]\[
2 \cdot 264.00 = 528.00
\][/tex]
So, the final result is [tex]\(528.00\)[/tex].
