Question

a robotic arm is at position $a(1,2,3)$ and needs to move to position $b(7,-4,5)$. what is the straight - line distance the arm needs to travel between these two points? $sqrt{20}$ units $sqrt{76}$ units $sqrt{80}$ units $sqrt{83}$ units

Explanation:

Step1: Recall the distance formula in 3D

The distance formula between two points \(A(x_1,y_1,z_1)\) and \(B(x_2,y_2,z_2)\) is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2+(z_2 - z_1)^2}\).
Here, \(x_1 = 1,y_1=2,z_1 = 3,x_2=7,y_2=-4,z_2 = 5\).

Step2: Substitute the values into the formula

First, find \((x_2 - x_1)\): \(7 - 1=6\).
Second, find \((y_2 - y_1)\): \(-4 - 2=-6\).
Third, find \((z_2 - z_1)\): \(5 - 3 = 2\).
Then, calculate \((x_2 - x_1)^2+(y_2 - y_1)^2+(z_2 - z_1)^2\):
\(6^2+(-6)^2 + 2^2=36 + 36+4\).
\(36+36 + 4=76\).

Step3: Calculate the distance

\(d=\sqrt{76}\).

Answer:

\(\sqrt{76}\) units