00:01
Hi there.
00:02
In this question we are given with the equation of the curve as r of t is equal to e power t, e power negative t 3t.
00:07
And we have to find the torsion and the formula is given to us.
00:10
And first we have to find the general formula for the torsion at a general point.
00:15
And then we have to find the tuition corresponding to t is equal to zero.
00:18
So let's see how we'll do that.
00:21
So first we'll find what is r dash of t here.
00:25
So r dash of t means we have to differentiate the components of r of t.
00:30
It is e power t, e power negative t, 3t.
00:33
So on differentiation, we'll be getting e power t, negative e power negative t, and three.
00:39
And r dash, double dash of t is equal to, again, differentiating, we'll be getting e power t, then comes e power negativity again, and then zero.
00:50
And our triple dash of t becomes e power t, negative e power t, negative e power negative, t, zero.
00:59
So these are r dash of t, r double dash of t, r triple dash of t.
01:03
Now we'll find r dash of t cross product with r double dash of t.
01:09
So that will be basically, we'll be having ijk, i'm writing it in this way.
01:16
And the r dash of t part will be the i the component is actually e power t here.
01:23
Here it's negative e power negative t and here it's 3 and here it is e power t, e power negative t and 0.
01:33
And taking the component, i mean, the determinant of this we'll be getting i multiplied by negative e power negative t multiplied by 0 minus 3 multiplied by e power negative t and minus j multiplied by j multiplied by e power t multiplied by 0 minus 3 multiplied by 3 multiplied by e power t and then plus k multiplied by e power t multiplied by e power negative t and minus minus e power minus of minus e power negativity multiplied by e power t and this will become i multiplied by 0 minus 3 e power negative t minus j multiplied by 0 minus 3 e power t plus k multiplied by e power 0 plus e power 0 and this will become minus 3 e power negative t i plus j i mean plus 3 e power t j plus 2k because e power 0 is 1.
02:56
So in component wise, we can write this as minus 3e power minus t, 3e power t and 2.
03:05
So this is what is r dash of t cross r double dash of t.
03:11
Now there in the denominator part, we are asked to find the modulus of r dash of t cross r double dash of t the whole square.
03:20
So that will be basically root of minus 3 e power minus t the whole square plus 3 e power t the whole square plus 2 square the whole square so the square and root get cancelled so we'll be having 9 e power negative 2 t plus 9 e power 2 t plus 4 and that will be equal to 9 multiplied by e power negative 2 t plus e power 2t plus 4.
03:57
So this will be the required modulus here.
04:03
And now we have to find our dash of t cross r double dash of t dot product with our triple dash of t...