Unit Tangent and Principal Unit Normal Vectors Calc IV Lab Karen Donnelly Saint Josepn's College

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 The following is to convince Maple that the variable t is only real-valued (not complex). QyYtSSdhc3N1bWVHNiI2IydJInRHRiVJJXJlYWxHRiUhIiItSSppbnRlcmZhY2VHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2Iy9JLHNob3dhc3N1bWVkR0YlIiIhRio=

Consider a position vector r(t). The velocity vector (also called the tangent vector) at point P = r(LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGPi8lL3N1YnNjcmlwdHNoaWZ0R0Y9RkBGPg==) is given by v(LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGPi8lL3N1YnNjcmlwdHNoaWZ0R0Y9RkBGPg==) = r'(LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGPi8lL3N1YnNjcmlwdHNoaWZ0R0Y9RkBGPg==). The direction of this vector direction is the direction of motion at time LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGPi8lL3N1YnNjcmlwdHNoaWZ0R0Y9RkBGPg==, and its magnitude || r'(LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGPi8lL3N1YnNjcmlwdHNoaWZ0R0Y9RkBGPg==)|| is the speed of the object at time LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGPi8lL3N1YnNjcmlwdHNoaWZ0R0Y9RkBGPg==. It also gives use the direction vector for the tangent line to the curve represented by r(t). The acceleration vector a for the position vector is given by a(t) = r''(t) = v'(t). It gives a measure of the rate of change of velocity -- i.e. rate at which the direction and speed of motion are changing. The unit tangent vector T(t) is the normalized "velocity" vector -- that is T(t) = r'(t)/|| r'(t)||. The unit tangent vector points in the direction of motion. (It of course also gives us the direction of the tangent line to the curve represented by r(t) The principal unit normal is defined to be N(t) = T'(t)/|| T'(t)|| It is orthogonal to the direction of motion, T(t), and points in the turning direction of the curve traced by r(t). (That is, it points towards the concave side of the curve). The acceleration vector a(t) can be decomposed into a linear combination of T(t) and N(t): a(t) = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIlRGJ0YyRjUvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRj1GQA==T(t) + LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIk5GJ0YyRjUvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRj1GQA==N(t), where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIlRGJ0YyRjUvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRj1GQA=== 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 =a.N . The scalars LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIlRGJ0YyRjUvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRj1GQA== and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIk5GJ0YyRjUvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRj1GQA== are called the tangential and normal components of acceleration. The VectorCalculus package with Maple contains built-in support for computing these vectors. JSFH

As an example consider a helix curve. We define with the PositionVector function so as to use certain options with PkkiUkc2Ii1JL1Bvc2l0aW9uVmVjdG9yR0YkNiM3JS1JIipHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YkSS9WZWN0b3JDYWxjdWx1c0c2JEYsSShfc3lzbGliR0YkNiQiIiUtSSRzaW5HRiQ2I0kidEdGJC1GKjYkRjMtSSRjb3NHRiRGNkY3 LUkzUGxvdFBvc2l0aW9uVmVjdG9yRzYiNiVJIlJHRiQvSSJ0R0YkOyIiIS1JIipHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YkSS9WZWN0b3JDYWxjdWx1c0c2JEYuSShfc3lzbGliR0YkNiQiIiVJI1BpR0YuL0ktY3VydmVvcHRpb25zR0YkNyMvSSVheGVzR0YkSSRib3hHRiQ= The following command will compute the derivative of R (unnormalized tangent vector). PkkiVkc2Ii1JLlRhbmdlbnRWZWN0b3JHRiQ2JEkiUkdGJEkidEdGJA== To get the unit tangent vector, we add the "normalized" option PkkiVEc2Ii1JLlRhbmdlbnRWZWN0b3JHRiQ2JUkiUkdGJEkidEdGJEkrbm9ybWFsaXplZEdGJA== Similarly the commands below compute first the "un-normalized", then "normalized" Principal Normal. LUkwUHJpbmNpcGFsTm9ybWFsRzYiNiRJIlJHRiRJInRHRiQ= PkkiTkc2Ii1JMFByaW5jaXBhbE5vcm1hbEdGJDYlSSJSR0YkSSJ0R0YkSStub3JtYWxpemVkR0Yk LUkzUGxvdFBvc2l0aW9uVmVjdG9yRzYiNidJIlJHRiQvSSJ0R0YkOyIiIS1JIipHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YkSS9WZWN0b3JDYWxjdWx1c0c2JEYuSShfc3lzbGliR0YkNiQiIiVJI1BpR0YuL0ktY3VydmVvcHRpb25zR0YkNyQvSSVheGVzR0YkSSZib3hlZEdGJC9JJmNvbG9yR0YkSSRyZWRHRiQvSSh0YW5nZW50R0YkSSV0cnVlR0YuL0knbm9ybWFsR0YuRkI= The acceleration vector a for this helix is given by a(t) = r''(t) = v'(t). PkkiQUc2Ii1JJWRpZmZHJSpwcm90ZWN0ZWRHNiRJIlZHRiRJInRHRiQ= The tangential and normal components of acceleration are calculated as follows: PkkjQVRHNiItSTBkZWxheURvdFByb2R1Y3RHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YkSSxUeXBlc2V0dGluZ0c2JEYoSShfc3lzbGliR0YkNiRJIkFHRiRJIlRHRiQ= QyQ+SSNBTkc2Ii1JMGRlbGF5RG90UHJvZHVjdEc2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHRiVJLFR5cGVzZXR0aW5nRzYkRilJKF9zeXNsaWJHRiU2JEkiQUdGJUkiTkdGJSIiIg==LUkpc2ltcGxpZnlHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0kjQU5HRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= JSFH

As an second example consider the knot curve representing the position vector r(t) = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobWZlbmNlZEdGJDYmLUYjNistSSNtaUdGJDYjUSFGJy1GIzYoLUYsNiQtRiM2KC1JI21uR0YkNiRRIjRGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSIrRidGPi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRy8lKXN0cmV0Y2h5R0ZHLyUqc3ltbWV0cmljR0ZHLyUobGFyZ2VvcEdGRy8lLm1vdmFibGVsaW1pdHNHRkcvJSdhY2NlbnRHRkcvJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZWLUYjNictRjE2JVEkc2luRicvJSdpdGFsaWNHRkdGPi1GQjYtUTAmQXBwbHlGdW5jdGlvbjtGJ0Y+RkVGSEZKRkxGTkZQRlIvRlVRJjAuMGVtRicvRlhGXm8tRiw2JC1GIzYlLUYxNiVRInRGJy9GaW5RJXRydWVGJy9GP1EnaXRhbGljRicvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnRj5GPkZbcEY+RjBGW3BGPkY+LUZCNi1RMSZJbnZpc2libGVUaW1lcztGJ0Y+RkVGSEZKRkxGTkZQRlJGXW9GX28tRiM2Jy1GMTYlUSRjb3NGJ0ZobkY+RmpuLUYsNiQtRiM2J0YwLUYjNictRjs2JFEiM0YnRj5GXnBGZG9GW3BGPkYwRltwRj5GPkZbcEY+RjBGW3BGPi1GQjYtUSIsRidGPkZFL0ZJRmhvRkpGTEZORlBGUkZdby9GWFEsMC4zMzMzMzMzZW1GJy1GIzYoRjZGXnAtRiM2J0ZlbkZqbkZmcEZbcEY+RjBGW3BGPkZfcS1GIzYnRmNwRmpuRmBvRltwRj5GMEZbcEY+Rj4vJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGW3BGPg== JSFHPkkiUkc2Ii1JL1Bvc2l0aW9uVmVjdG9yR0YkNiM3JS1JIipHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YkSS9WZWN0b3JDYWxjdWx1c0c2JEYsSShfc3lzbGliR0YkNiQtSSIrR0YrNiQiIiUtSSRzaW5HRjA2I0kidEdGJC1JJGNvc0dGMDYjLUYqNiQiIiRGOi1GKjYkRjMtRjhGPS1GPEY5 An plot of this curve from t=0 to 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, with some unit tangent vectors and unit normal vectors to the curve: LUkzUGxvdFBvc2l0aW9uVmVjdG9yRzYiNidJIlJHRiQvSSJ0R0YkOyIiIS1JIipHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YkSS9WZWN0b3JDYWxjdWx1c0c2JEYuSShfc3lzbGliR0YkNiQiIiNJI1BpR0YuL0ktY3VydmVvcHRpb25zR0YkNyUvSSVheGVzR0YkSSZib3hlZEdGJC9JJmNvbG9yR0YkSSRyZWRHRiQvSSpudW1wb2ludHNHRiQiJCsiL0kodGFuZ2VudEdGJEkldHJ1ZUdGLi9JJ25vcm1hbEdGLkZF We can see we get a really complicated expression for the unit tangent vector and principal unit normal vector: PkkiVkc2Ii1JLlRhbmdlbnRWZWN0b3JHRiQ2JEkiUkdGJEkidEdGJA== PkkiVEc2Ii1JLlRhbmdlbnRWZWN0b3JHRiQ2JUkiUkdGJEkidEdGJEkrbm9ybWFsaXplZEdGJA== LUkwUHJpbmNpcGFsTm9ybWFsRzYiNiRJIlJHRiRJInRHRiQ= PkkiTkc2Ii1JMFByaW5jaXBhbE5vcm1hbEdGJDYlSSJSR0YkSSJ0R0YkSStub3JtYWxpemVkR0Yk LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

(This exercise is from Larson, Section 12.4) Find 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 for the vector valued function 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 at the point t =2. Verify the relationship 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. Do this by completing the commands below, documenting each with a text cell explaining what you are calculating. PkkiUkc2Ii1JL1Bvc2l0aW9uVmVjdG9yR0YkNiM3JUkidEdGJC1JIipHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YkSS9WZWN0b3JDYWxjdWx1c0c2JEYtSShfc3lzbGliR0YkNiQiIiQqJEYpIiIjLUYrNiRGNSMiIiJGNg== 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 LUkzUGxvdFBvc2l0aW9uVmVjdG9yRzYiNiVJIlJHRiQvSSJ0R0YkOyIiISIiJS9JLWN1cnZlb3B0aW9uc0dGJDckL0klYXhlc0dGJEkmYm94ZWRHRiQvSSZjb2xvckdGJEkkcmVkR0YkJSFH PkkiVkc2Ii1JJWRpZmZHJSpwcm90ZWN0ZWRHNiRJIlJHRiRJInRHRiQ= PkkjVjFHNiItSSZldmFsZkclKnByb3RlY3RlZEc2Iy1JJXN1YnNHRic2JC9JInRHRiQiIiNJIlZHRiQ= PkkiQUc2Ii1JJWRpZmZHJSpwcm90ZWN0ZWRHNiRJIlZHRiRJInRHRiQ= PkkjQTFHNiItSSVzdWJzRyUqcHJvdGVjdGVkRzYkL0kidEdGJCIiI0kiQUdGJA== PkkiVEc2Ii1JLlRhbmdlbnRWZWN0b3JHRiQ2JUkiUkdGJEkidEdGJEkrbm9ybWFsaXplZEdGJA== Or alternatively PkkiVEc2Ii1JIipHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YkSS9WZWN0b3JDYWxjdWx1c0c2JEYoSShfc3lzbGliR0YkNiRJIlZHRiQqJC1JJXNxcnRHRiw2Iy1JMGRlbGF5RG90UHJvZHVjdEc2JEYoL0YqSSxUeXBlc2V0dGluZ0dGLDYkRi9GLyEiIg== QyQ+SSNUMUc2Ii1JJXN1YnNHJSpwcm90ZWN0ZWRHNiQvSSJ0R0YlIiIjSSJUR0YlIiIiLUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiNJIiVHNiI= PkkiTkc2Ii1JMFByaW5jaXBhbE5vcm1hbEdGJDYlSSJSR0YkSSJ0R0YkSStub3JtYWxpemVkR0Yk Or, alternatively QyU+SSNkVEc2Ii1JKXNpbXBsaWZ5RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMtSSVkaWZmR0YpNiRJIlRHRiVJInRHRiUiIiI+SSJOR0YlLUYnNiMtSSIqRzYkRikvSSttb2R1bGVuYW1lR0YlSS9WZWN0b3JDYWxjdWx1c0dGKDYkRiQqJC1JJXNxcnRHRig2Iy1JMGRlbGF5RG90UHJvZHVjdEc2JEYpL0Y6SSxUeXBlc2V0dGluZ0dGKDYkRiRGJCEiIg== PkkjTjFHNiItSSVzdWJzRyUqcHJvdGVjdGVkRzYkL0kidEdGJCIiI0kiTkdGJA== QyU+SSNhVEc2Ii1JMGRlbGF5RG90UHJvZHVjdEc2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHRiVJLFR5cGVzZXR0aW5nRzYkRilJKF9zeXNsaWJHRiU2JEkjQTFHRiVJI1QxR0YlIiIiLUkmZXZhbGZHRik2I0kjYU5HRiU= QyY+SSNhTkc2Ii1JMGRlbGF5RG90UHJvZHVjdEc2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHRiVJLFR5cGVzZXR0aW5nRzYkRilJKF9zeXNsaWJHRiU2JEkjQTFHRiVJI04xR0YlIiIiLUkmZXZhbGZHRik2I0YkRjI= Verify the linear decomposition of acceleration: a = 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 at the point t = 2 LUkiK0c2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJL1ZlY3RvckNhbGN1bHVzRzYkRiVJKF9zeXNsaWJHRig2JC1JIipHRiQ2JEkjYVRHRihJI1QxR0YoLUYuNiRJI2FOR0YoSSNOMUdGKA== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

From the VectorCalculus package: PositionVector PlotVector PlotPositionVector TangentVector PrincipalNormal