To determine the magnitude of the resultant displacement of the spider, we need to consider its movements in three dimensions:
1. Movement due north: The spider crawls 1 meter north. We can denote this as [tex]\(x = 1\)[/tex] meter.
2. Movement due east: The spider then crawls 2 meters east. We can denote this as [tex]\(y = 2\)[/tex] meters.
3. Vertical movement upwards: Finally, the spider climbs 3 meters vertically upwards. We can denote this as [tex]\(z = 3\)[/tex] meters.
The resultant displacement is a vector that combines these three individual movements. To find the magnitude of this resultant displacement vector, we use the three-dimensional version of the Pythagorean theorem:
[tex]\[
\text{Resultant Displacement} = \sqrt{x^2 + y^2 + z^2}
\][/tex]
Now plug in the given values:
[tex]\[
\text{Resultant Displacement} = \sqrt{(1)^2 + (2)^2 + (3)^2}
\][/tex]
Performing each of the calculations inside the square root:
[tex]\[
1^2 = 1
\][/tex]
[tex]\[
2^2 = 4
\][/tex]
[tex]\[
3^2 = 9
\][/tex]
Now, sum these squared values:
[tex]\[
1 + 4 + 9 = 14
\][/tex]
Finally, take the square root of the sum:
[tex]\[
\sqrt{14} \approx 3.7416573867739413
\][/tex]
Thus, the magnitude of the resultant displacement of the spider is approximately [tex]\(3.7416573867739413\)[/tex] meters.
