A force of (2î - 4ĵ + 2k̂)N acts at a point (3î + 2ĵ -4k̂ ) metre from the origin. The magnitude of torque is

The correct answer is 24.4 N-m.

Step-by-Step Explanation:

Torque ($\tau$) is defined as the cross product of the position vector ($r$) and the force vector ($F$):
$\tau =r\times F$

Given:
Force vector, $F=2\mathrm{î}-4\mathrm{ĵ}+2\mathrm{k̂}$
Position vector, $r=3\mathrm{î}+2\mathrm{ĵ}-4\mathrm{k̂}$

We calculate the cross product using the determinant of a 3×3 matrix:
$\tau =\left|\begin{array}{ccc}\mathrm{î}& \mathrm{ĵ}& \mathrm{k̂}\\ 3& 2& -4\\ 2& -4& 2\end{array}\right|$

Expanding the determinant along the first row:
$\tau =\mathrm{î}\left[\right(2\left)\right(2)-(-4\left)\right(-4\left)\right]-\mathrm{ĵ}\left[\right(3\left)\right(2)-(-4\left)\right(2\left)\right]+\mathrm{k̂}\left[\right(3\left)\right(-4)-(2\left)\right(2\left)\right]$

Simplify the terms inside the brackets:
$\tau =\mathrm{î}[4-16]-\mathrm{ĵ}[6-(-8\left)\right]+\mathrm{k̂}[-12-4]$

$\tau =-12\mathrm{î}-14\mathrm{ĵ}-16\mathrm{k̂}$

Now, find the magnitude of the torque vector ($\left|\tau \right|$):
$\left|\tau \right|=\sqrt{{(-12)}^2+{(-14)}^2+{(-16)}^2}$

Calculate the squares:
$\left|\tau \right|=\sqrt{144+196+256}$

$\left|\tau \right|=\sqrt{596}\approx 24.41\text{N-m}$

Rounding to one decimal place, the magnitude of the torque is 24.4 N-m.