The image presents a set of problems. We need to solve problem 4, which involves vector calculations. Given the vectors: $\vec{A} = -8\hat{k} - 17\hat{j}$ $\vec{B} = 620\hat{i} - 35\hat{k}$ $\vec{C} = -50\hat{i} + 43\hat{j} + 76\hat{k}$ We need to calculate: a) $\vec{C} \times \vec{A}$ b) $\vec{A} \times \vec{B} \cdot \vec{C}$ c) A vector of magnitude 60 in the direction of the resultant vector $\vec{A}+\vec{B}+\vec{C}$

1. Problem Description

The image presents a set of problems. We need to solve problem 4, which involves vector calculations.

Given the vectors:

$\vec{A} = -8\hat{k} - 17\hat{j}$

$\vec{B} = 620\hat{i} - 35\hat{k}$

$\vec{C} = -50\hat{i} + 43\hat{j} + 76\hat{k}$

We need to calculate:

a) $\vec{C} \times \vec{A}$

b) $\vec{A} \times \vec{B} \cdot \vec{C}$

c) A vector of magnitude 60 in the direction of the resultant vector $\vec{A}+\vec{B}+\vec{C}$

2. Solution Steps

a) $\vec{C} \times \vec{A}$

$\vec{C} \times \vec{A} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ -50 & 43 & 76 \\ 0 & -17 & -8 \end{vmatrix} = \hat{i}(43(-8) - 76(-17)) - \hat{j}((-50)(-8) - 76(0)) + \hat{k}((-50)(-17) - 43(0))$

$= \hat{i}(-344 + 1292) - \hat{j}(400 - 0) + \hat{k}(850 - 0) = 948\hat{i} - 400\hat{j} + 850\hat{k}$

b) $\vec{A} \times \vec{B} \cdot \vec{C}$

First, calculate $\vec{A} \times \vec{B}$:

$\vec{A} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 0 & -17 & -8 \\ 620 & 0 & -35 \end{vmatrix} = \hat{i}((-17)(-35) - (-8)(0)) - \hat{j}((0)(-35) - (-8)(620)) + \hat{k}((0)(0) - (-17)(620))$

$= \hat{i}(595 - 0) - \hat{j}(0 + 4960) + \hat{k}(0 + 10540) = 595\hat{i} - 4960\hat{j} + 10540\hat{k}$

Now, calculate the dot product $(\vec{A} \times \vec{B}) \cdot \vec{C}$:

$(\vec{A} \times \vec{B}) \cdot \vec{C} = (595\hat{i} - 4960\hat{j} + 10540\hat{k}) \cdot (-50\hat{i} + 43\hat{j} + 76\hat{k})$

$= (595)(-50) + (-4960)(43) + (10540)(76) = -29750 - 213280 + 801040 = 558010$

c) A vector of magnitude 60 in the direction of the resultant vector $\vec{A}+\vec{B}+\vec{C}$

First, calculate $\vec{A} + \vec{B} + \vec{C}$:

$\vec{A} + \vec{B} + \vec{C} = (0\hat{i} - 17\hat{j} - 8\hat{k}) + (620\hat{i} + 0\hat{j} - 35\hat{k}) + (-50\hat{i} + 43\hat{j} + 76\hat{k})$

$= (0 + 620 - 50)\hat{i} + (-17 + 0 + 43)\hat{j} + (-8 - 35 + 76)\hat{k} = 570\hat{i} + 26\hat{j} + 33\hat{k}$

The magnitude of the resultant vector $\vec{R}$ is $|\vec{R}| = \sqrt{(570)^2 + (26)^2 + (33)^2} = \sqrt{324900 + 676 + 1089} = \sqrt{326665} \approx 571.546$

The unit vector in the direction of $\vec{R}$ is $\hat{u} = \frac{\vec{R}}{|\vec{R}|} = \frac{570\hat{i} + 26\hat{j} + 33\hat{k}}{571.546} \approx 0.997\hat{i} + 0.045\hat{j} + 0.058\hat{k}$

The vector with magnitude 60 in the direction of $\vec{R}$ is $60\hat{u} = 60(0.997\hat{i} + 0.045\hat{j} + 0.058\hat{k}) \approx 59.82\hat{i} + 2.7\hat{j} + 3.48\hat{k}$

3. Final Answer

a) $\vec{C} \times \vec{A} = 948\hat{i} - 400\hat{j} + 850\hat{k}$

b) $\vec{A} \times \vec{B} \cdot \vec{C} = 558010$

c) $59.82\hat{i} + 2.7\hat{j} + 3.48\hat{k}$