Explanation Given information: r ( t ) = ( cos t ) i + ( sin t ) j − k (1) Formula used: Write the general expression to calculate the velocity. v = d r d t (2) Here, d r d t is the rate of change in displacement with respect to time. Calculation: Substitute π 4 for t in equation (1). r ( π 4 ) = ( cos π 4 ) i + ( sin π 4 ) j − k = 2 2 i + 2 2 j − k For r at t = π 4 , the point P is ( 2 2 , 2 2 , − 1 ) . Substitute equation (1) in equation (2). v = d d t ( ( cos t ) i + ( sin t ) j − k ) v = ( − sin t ) i + ( cos t ) j (3) Take magnitude on both sides of the above equation. | v | = sin 2 t + cos 2 t | v | = 1 { ∵ sin 2 θ + cos 2 θ = 1 } (4) The tangential vector T is calculated as follows. T = v | v | (5) Substitute equation (3) and (4) in equation (5). T = ( − sin t ) i + ( cos t ) j 1 T = ( − sin t ) i + ( cos t ) j (6) At t = π 4 , the equation (6) becomes, T ( π 4 ) = ( − sin π 4 ) i + ( cos π 4 ) j = − 2 2 i + 2 2 j Differentiate the equation (6) with respect to time. d T d t = ( − cos t ) i + ( − sin t ) j (7) Take magnitude on both sides of the above equation. | d T d t | = ( − cos t ) 2 + ( − sin t ) 2 = cos 2 t + sin 2 t | d T d t | = 1 { ∵ sin 2 θ + cos 2 θ = 1 } (8) The normal vector N is calculated as follows. N = d T d t | d T d t | (9) Substitute equation (7) and (8) in equation (9). N = ( − cos t ) i + ( − sin t ) j 1 N = ( − cos t ) i + ( − sin t ) j (10) At t = π 4 , the equation (10) becomes, N ( π 4 ) = ( − cos π 4 ) i + ( − sin π 4 ) j = − 2 2 i − 2 2 j The binormal vector B is calculated as follows

Thomas' Calculus Format: Unbound (saleable) With Access Card

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ISBN: 9780134768762

Author: Hass, Joel R.^heil, Christopher D.^weir, Maurice D.

Publisher: Prentice Hall

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Chapter 13.5, Problem 7E

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Calculate the value of r, T, N and B at t=π4. Also find the equations for the osculating, normal, and rectifying planes for given vector r(t)=(cost)i+(sint)j−k, t=π4.

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Thomas' Calculus Format: Unbound (saleable) With Access Card

Ch. 13.1 - In Exercises 1–4, find the given limits. 1. Ch. 13.1 - In Exercises 1–4, find the given limits. 2. Ch. 13.1 - In Exercises 1–4, find the given limits. 3. Ch. 13.1 - In Exercises 1–4, find the given limits. 4. Ch. 13.1 - Motion in the Plane In Exercises 5–8, r(t) is the...Ch. 13.1 - Motion in the Plane In Exercises 5–8, r(t) is the...Ch. 13.1 - Prob. 7E Ch. 13.1 - Prob. 8E Ch. 13.1 - Prob. 9E Ch. 13.1 - Prob. 10E

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Calculate the value of r , T , N and B at t = π 4 . Also find the equations for the osculating, normal, and rectifying planes for given vector r ( t ) = ( cos t ) i + ( sin t ) j − k , t = π 4 .

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Calculate the value of r, T, N and B at t=π4. Also find the equations for the osculating, normal, and rectifying planes for given vector r(t)=(cost)i+(sint)j−k, t=π4.

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