Find T, N, and B for the given space curve. r(t) = (1 + 3 sin 4/3 t)i + (10 + 3

Question

Find T, N, and B for the given space curve. r(t) = (1 + 3 sin 4/3 t)i + (10 + 3 cos 4/3 t)j + 3tk T = 4/5 (cos 1.333t)i - 4/5 (sin 1.333t)j; N = (-sin 1.333t)I - (cos 1.333t)j; B = 3/5 (cos 1.333)i - 3/5 (sin 1.333t)j - 4/5 k T = 4/5 (cos 1.333t)i - 4/5 (sin 1.333t)j + 3/5 k;N = (-sin 1.333t)i - (cos 1.333t)j; B = - 4/5 k T = 4/5 (sin 1.333t)i - 4/5 (cos 1.333t)j; N = (-sin 1.333t)i - (cos 1.333t)j; B = 3/5 (cos 1.333t)i - 3/5 (sin 1.333t)j - 4/5 k T = 4/5 (cos 1.333t)i - 4/5 (sin 1.333t)j + 3/5 k; N = (-sin 1.333t)i - (cos 1.333t)j; B = 3/5 (cos 1.333t)i - 3/5(sin 1.333t)j - 4/5 k

Explanation / Answer

given r(t)=(1+3sin(4/3)t)i +(10+ 3cos(4/3)t)j +3tk

r'(t)=(0+3(4/3)cos(4/3)t)i +(0- 3(4/3)sin(4/3)t)j +3k

r'(t)=(4cos(4/3)t)i -(4sin(4/3)t)j +3k

|r'(t)|=[(4cos(4/3)t)2+(-4sin(4/3)t)2 +32]

|r'(t)|=[(4)2 +32]=5

T(t)=r'(t)/|r'(t)|

T(t)=((4/5)cos(4/3)t)i -((4/5)sin(4/3)t)j +(3/5)k

T(t)=((4/5)cos(1.333)t)i -((4/5)sin(1.333)t)j +(3/5)k

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T'(t)=-((4/5)(4/3)sin(4/3)t)i -((4/5)(4/3)cos(4/3)t)j +(0)k

T'(t)=-((16/15)sin(4/3)t)i -((16/15)cos(4/3)t)j +(0)k

|T'(t)|=[(-((16/15)sin(4/3)t))2+(-((16/15)cos(4/3))2]

|T'(t)|=16/15

N(t)=T'(t)/|T'(t)|

N(t)=-(sin(4/3)t)i -(cos(4/3)t)j +(0)k

N(t)=-(sin(1.333)t)i -(cos(1.333)t)j +(0)k

...............................................................

T(t)=((4/5)cos(1.333)t)i -((4/5)sin(1.333)t)j +(3/5)k

N(t)=-(sin(1.333)t)i -(cos(1.333)t)j +(0)k

B(t) =T(t)xN(t)

B(t) =((3/5)cos(1.333)t)i-((3/5)sin(1.333)t)j + (-((4/5)cos2(1.333)t)2-((4/5)sin2(1.333)t)2)k

B(t) =((3/5)cos(1.333)t)i-((3/5)sin(1.333)t)j - (4/5)k

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