Answer
a) $\lt 7, 1, 5\gt$
b) $8.66 \ ft$
Work Step by Step
a) The position of the robot hand can be computed by adding the vectors of its arms.
That is,
$d=a+b+c = \lt 2,3,4 \gt +\lt 1,-1, 3 \gt +\lt 4, -1, -2 \gt =\lt 2+1+4, 3-1 -1, 4+3-2 \gt=\lt 7, 1, 5\gt$
b) The distance of the hand from the origin is equal to the magnitude of vector $d$.
$||d||=\sqrt {(7^2)+(1^2)+(5^2)} \\=\sqrt {75} \\ \approx 8.66 \ ft$
