To determine the magnitude of the displacement of the particle moving from point [tex]\( A(1, -2, -3) \)[/tex] to point [tex]\( B(3, 4, 5) \)[/tex], we perform the following steps:
1. Calculate the components of the displacement vector [tex]\(\vec{d}\)[/tex]:
- The displacement vector [tex]\(\vec{d}\)[/tex] has components given by the differences between the coordinates of point B and point A.
- Determine the differences along the x, y, and z axes:
[tex]\[
d_x = x_B - x_A = 3 - 1 = 2
\][/tex]
[tex]\[
d_y = y_B - y_A = 4 - (-2) = 4 + 2 = 6
\][/tex]
[tex]\[
d_z = z_B - z_A = 5 - (-3) = 5 + 3 = 8
\][/tex]
2. Determine the magnitude of the displacement vector [tex]\(\| \vec{d} \|\)[/tex]:
- The magnitude of the displacement vector [tex]\(\| \vec{d} \|\)[/tex] is given by the Euclidean distance formula for three-dimensional space:
[tex]\[
\| \vec{d} \| = \sqrt{d_x^2 + d_y^2 + d_z^2}
\][/tex]
- Plugging in the calculated components:
[tex]\[
\| \vec{d} \| = \sqrt{2^2 + 6^2 + 8^2}
\][/tex]
- Calculate each term inside the square root:
[tex]\[
2^2 = 4, \quad 6^2 = 36, \quad 8^2 = 64
\][/tex]
- Sum these squares:
[tex]\[
4 + 36 + 64 = 104
\][/tex]
- Thus,
[tex]\[
\| \vec{d} \| = \sqrt{104}
\][/tex]
- Calculating the square root:
[tex]\[
\sqrt{104} \approx 10.198
\][/tex]
Therefore, the magnitude of the displacement of the particle is approximately [tex]\(10.198 \, m\)[/tex].
Thus, the correct answer is none of the options provided [tex]\((a) 7.1 \, m, (b) 3.7 \, m, (c) 3.4 \, m, (d) 8.5 \, m\)[/tex]. If there had been an option for [tex]\(10.2 \, m\)[/tex], it would have been a closer match.
