To determine the magnitude of the resultant force acting on the box when two forces [tex]\( \mathbf{F_1} = 2\mathbf{i} - 4\mathbf{j} + \mathbf{k} \)[/tex] and [tex]\( \mathbf{F_2} = 4\mathbf{i} - \mathbf{j} - 3\mathbf{k} \)[/tex] are applied, follow these steps:
1. Find the resultant force [tex]\(\mathbf{F_R}\)[/tex]
The resultant force [tex]\(\mathbf{F_R}\)[/tex] is found by vector addition of [tex]\(\mathbf{F_1}\)[/tex] and [tex]\(\mathbf{F_2}\)[/tex]:
[tex]\[
\mathbf{F_R} = \mathbf{F_1} + \mathbf{F_2}
\][/tex]
Given:
[tex]\[
\mathbf{F_1} = 2\mathbf{i} - 4\mathbf{j} + \mathbf{k}
\][/tex]
[tex]\[
\mathbf{F_2} = 4\mathbf{i} - \mathbf{j} - 3\mathbf{k}
\][/tex]
Now, add the corresponding components:
- For the [tex]\(\mathbf{i}\)[/tex] component:
[tex]\[
2 + 4 = 6
\][/tex]
- For the [tex]\(\mathbf{j}\)[/tex] component:
[tex]\[
-4 - 1 = -5
\][/tex]
- For the [tex]\(\mathbf{k}\)[/tex] component:
[tex]\[
1 - 3 = -2
\][/tex]
So, the resultant force [tex]\(\mathbf{F_R}\)[/tex] is:
[tex]\[
\mathbf{F_R} = 6\mathbf{i} - 5\mathbf{j} - 2\mathbf{k}
\][/tex]
2. Calculate the magnitude of the resultant force [tex]\(\mathbf{F_R}\)[/tex]
The magnitude of a vector [tex]\(\mathbf{F_R} = a\mathbf{i} + b\mathbf{j} + c\mathbf{k} \)[/tex] is given by:
[tex]\[
|\mathbf{F_R}| = \sqrt{a^2 + b^2 + c^2}
\][/tex]
Here, [tex]\( a = 6 \)[/tex], [tex]\( b = -5 \)[/tex], and [tex]\( c = -2 \)[/tex]:
[tex]\[
|\mathbf{F_R}| = \sqrt{6^2 + (-5)^2 + (-2)^2}
\][/tex]
Calculate each term inside the square root:
[tex]\[
| \mathbf{F_R} | = \sqrt{6^2 + (-5)^2 + (-2)^2}
\][/tex]
[tex]\[
= \sqrt{36 + 25 + 4}
\][/tex]
[tex]\[
= \sqrt{65}
\][/tex]
[tex]\[
\approx 8.062
\][/tex]
Thus, the magnitude of the resultant force acting on the box is approximately [tex]\( 8.06 \)[/tex] N.
3. Matching with given options
The options provided were:
[tex]\[
[A] 12 N
\][/tex]
[tex]\[
[B] 7 N
\][/tex]
[tex]\[
[C] 16 N
\][/tex]
[tex]\[
[D] -1 N
\][/tex]
Since none of these answers are exactly correct and based on our calculation, the closest would be [tex]\(8.06 N\)[/tex], which isn't provided in the given options. However, this value closest matches the typical precision for such a problem, though there seems to be a discrepancy in the provided answers.
Thus, the correct magnitude, [tex]\( 8.06 N \)[/tex], isn't directly shown, but based on proper workings through the provided steps, this is the result we found.
