Vectors in Maple Calc III Lab The VectorCalculus Package When studying vectors using Maple it is useful to employ procedures that are part of the Vector Calculus package. This package is made available by the with(Student[VectorCalculus]) command. (You can also select it with the pull down menu Tools-Load Package-VectorCalculus. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2Iy1GLDYlUShyZXN0YXJ0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4yNzc3Nzc4ZW1GJw==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 We execute the next command to load the plots library so that we can plot vectors as arrows with the arrow command from this library. LUkld2l0aEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSZwbG90c0dGJw==

1. Enter the following vectors u and v with standard components 1, 2, and 4, -1. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNictRiw2JVEjdTJGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSM6PUYnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZTLUkobWZlbmNlZEdGJDYmLUYjNictSSNtbkdGJDYkUSIxRidGQC1GPTYtUSIsRidGQEZCL0ZGRjhGR0ZJRktGTUZPL0ZSUSYwLjBlbUYnL0ZVUSwwLjMzMzMzMzNlbUYnLUZmbjYkUSIyRidGQC8lK2V4ZWN1dGFibGVHRkRGQEZALyUlb3BlbkdRMyZMZWZ0QW5nbGVCcmFja2V0O0YnLyUmY2xvc2VHUTQmUmlnaHRBbmdsZUJyYWNrZXQ7RidGZG9GQEYrRmRvRkBGK0Zkb0ZA PkkjdjJHNiItSSQ8LD5HRiQ2JCIiJS1JIi1HNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YkSS9WZWN0b3JDYWxjdWx1c0c2JUYsRiwvRi5JKFN0dWRlbnRHNiRGLEkoX3N5c2xpYkdGJDYjIiIi 2. The vector w =<x,y> with variable components is entered in the same way: Pkkid0c2Ii1JJDwsPkdGJDYkSSJ4R0YkSSJ5R0Yk Creating vectors in 3-space QyQ+SSN1M0c2Ii1JJDwsPkdGJTYlIiIiIiIjIiIkRik=PkkjdjNHNiItSSQ8LD5HRiQ2JS1JIi1HNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YkSS9WZWN0b3JDYWxjdWx1c0c2JUYrRisvRi1JKFN0dWRlbnRHNiRGK0koX3N5c2xpYkdGJDYjIiIjIiIiRjU=

Adding two vectors LUkiK0c2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJL1ZlY3RvckNhbGN1bHVzRzYlRiVGJS9GJ0koU3R1ZGVudEc2JEYlSShfc3lzbGliR0YoNiRJI3UyR0YoSSN2MkdGKA== LUkiK0c2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJL1ZlY3RvckNhbGN1bHVzRzYlRiVGJS9GJ0koU3R1ZGVudEc2JEYlSShfc3lzbGliR0YoNiRJI3UzR0YoSSN2M0dGKA== Multiplying by scalar: LUkiKkc2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJL1ZlY3RvckNhbGN1bHVzRzYlRiVGJS9GJ0koU3R1ZGVudEc2JEYlSShfc3lzbGliR0YoNiQiIiNJI3UyR0Yo LUkiLUc2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJL1ZlY3RvckNhbGN1bHVzRzYlRiVGJS9GJ0koU3R1ZGVudEc2JEYlSShfc3lzbGliR0YoNiMtSSIqR0YkNiQiIiRJI3UzR0Yo

QyQ+SSNhMUc2Ii1JJmFycm93R0YlNiVJI3UyR0YlL0kmd2lkdGhHRiUkIiIiISIiL0kmY29sb3JHRiVJJmdyZWVuR0YlRi4= QyQ+SSNhMkc2Ii1JJmFycm93R0YlNiVJI3YyR0YlL0kmd2lkdGhHRiUkIiIiISIiL0kmY29sb3JHRiVJJHJlZEdGJUYu QyQ+SSNhM0c2Ii1JJmFycm93R0YlNiUtSSIrRzYkJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUdGJUkvVmVjdG9yQ2FsY3VsdXNHNiVGLEYsL0YuSShTdHVkZW50RzYkRixJKF9zeXNsaWJHRiU2JEkjdTJHRiVJI3YyR0YlL0kmd2lkdGhHRiUkIiIiISIiL0kmY29sb3JHRiVJJWJsdWVHRiVGPA== LUkoZGlzcGxheUc2IjYnSSNhMUdGJEkjYTJHRiRJI2EzR0YkL0koc2NhbGluZ0dGJEksQ09OU1RSQUlORURHRiQvSSVheGVzR0YkSSZib3hlZEdGJA== If we want to plot a vector that is NOT in the standard position (i.e. initial point other than at the origin), we modify the arrow command, giving the coordinates to the terminal point first -- i.e. to plot the vector u2 = <1,2> with its tail at the tip of v2 = <4,-1>. LUkmYXJyb3dHNiI2Jy1JJDwsPkdGJDYkIiIlLUkiLUc2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHRiRJL1ZlY3RvckNhbGN1bHVzRzYkRi1JKF9zeXNsaWJHRiQ2IyIiIi1GJzYkRjQiIiMvSSZ3aWR0aEdGJCRGNCEiIi9JJmNvbG9yR0YkSSZncmVlbkdGJC9JJXZpZXdHRiQ3JDstRis2I0Y3IiInO0ZDRik= We can use these commands together to demonstrate the geometric interpretation of vector addition and also that vector addition is commutative. QyQ+SSNhMUc2Ii1JJmFycm93R0YlNiVJI3UyR0YlL0kmd2lkdGhHRiUkIiIiISIiL0kmY29sb3JHRiVJJmdyZWVuR0YlRi4= QyQ+SSNhMkc2Ii1JJmFycm93R0YlNiVJI3YyR0YlL0kmd2lkdGhHRiUkIiIiISIiL0kmY29sb3JHRiVJJHJlZEdGJUYu QyQ+SSNhM0c2Ii1JJmFycm93R0YlNiUtSSIrRzYkJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUdGJUkvVmVjdG9yQ2FsY3VsdXNHNiVGLEYsL0YuSShTdHVkZW50RzYkRixJKF9zeXNsaWJHRiU2JEkjdTJHRiVJI3YyR0YlL0kmd2lkdGhHRiUkIiIiISIiL0kmY29sb3JHRiVJJWJsdWVHRiVGPA== QyQ+SSNhNEc2Ii1JJmFycm93R0YlNiZJI3YyR0YlSSN1MkdGJS9JJndpZHRoR0YlJCIiIiEiIi9JJmNvbG9yR0YlSSd5ZWxsb3dHRiVGLw== QyQ+SSNhNUc2Ii1JJmFycm93R0YlNiZJI3UyR0YlSSN2MkdGJS9JJndpZHRoR0YlJCIiIiEiIi9JJmNvbG9yR0YlSShtYWdlbnRhR0YlRi8= LUkoZGlzcGxheUc2IjYqSSNhMUdGJEkjYTJHRiRJI2EzR0YkSSNhNEdGJEkjYTVHRiQvSShzY2FsaW5nR0YkSSxDT05TVFJBSU5FREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJC9JJWF4ZXNHRiRJJ25vcm1hbEdGLy9JJXZpZXdHRiQ3JDstSSItRzYkRi8vSSttb2R1bGVuYW1lR0YkSS9WZWN0b3JDYWxjdWx1c0c2JUYvRi8vRjxJKFN0dWRlbnRHRi42IyIiIyIiJztGOCIiJQ== QyQ+SSNjMUc2Ii1JJmFycm93R0YlNiVJI3YzR0YlL0kmd2lkdGhHRiUkIiIiISIiL0kmY29sb3JHRiVJJmdyZWVuR0YlRi4= QyQ+SSNjMkc2Ii1JJmFycm93R0YlNiUtSSItRzYkJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUdGJUkvVmVjdG9yQ2FsY3VsdXNHNiVGLEYsL0YuSShTdHVkZW50RzYkRixJKF9zeXNsaWJHRiU2Iy1JIipHRis2JCQiIzohIiJJI3YzR0YlL0kmd2lkdGhHRiUkIiIiRjsvSSZjb2xvckdGJUkkcmVkR0YlRjs= QyQ+SSNjM0c2Ii1JJmFycm93R0YlNiUtSSIqRzYkJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUdGJUkvVmVjdG9yQ2FsY3VsdXNHNiVGLEYsL0YuSShTdHVkZW50RzYkRixJKF9zeXNsaWJHRiU2JCQiIiYhIiJJI3YzR0YlL0kmd2lkdGhHRiUkIiIiRjgvSSZjb2xvckdGJUkneWVsbG93R0YlRjg= LUkoZGlzcGxheUc2IjYmPCVJI2MxR0YkSSNjMkdGJEkjYzNHRiQvSSVheGVzR0YkSSdub3JtYWxHJSpwcm90ZWN0ZWRHL0koc2NhbGluZ0dGJEksY29uc3RyYWluZWRHRiQvSSV2aWV3R0YkNyU7LUkiLUc2JEYtL0krbW9kdWxlbmFtZUdGJEkvVmVjdG9yQ2FsY3VsdXNHNiVGLUYtL0Y5SShTdHVkZW50RzYkRi1JKF9zeXNsaWJHRiQ2IyIiJUZBRjRGNA== The geometry also holds for vectors in three space. QyQ+SSNiMUc2Ii1JJmFycm93R0YlNiVJI3UzR0YlL0kmd2lkdGhHRiUkIiIiISIiL0kmY29sb3JHRiVJJmdyZWVuR0YlRi4= QyQ+SSNiMkc2Ii1JJmFycm93R0YlNiVJI3YzR0YlL0kmd2lkdGhHRiUkIiIiISIiL0kmY29sb3JHRiVJJHJlZEdGJUYu QyQ+SSNiM0c2Ii1JJmFycm93R0YlNiUtSSIrRzYkJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUdGJUkvVmVjdG9yQ2FsY3VsdXNHNiVGLEYsL0YuSShTdHVkZW50RzYkRixJKF9zeXNsaWJHRiU2JEkjdTNHRiVJI3YzR0YlL0kmd2lkdGhHRiUkIiIiISIiL0kmY29sb3JHRiVJJWJsdWVHRiVGPA== QyQ+SSNiNEc2Ii1JJmFycm93R0YlNiZJI3YzR0YlSSN1M0dGJS9JJndpZHRoR0YlJCIiIiEiIi9JJmNvbG9yR0YlSSd5ZWxsb3dHRiVGLw== QyQ+SSNiNUc2Ii1JJmFycm93R0YlNiZJI3UzR0YlSSN2M0dGJS9JJndpZHRoR0YlJCIiIiEiIi9JJmNvbG9yR0YlSShtYWdlbnRhR0YlRi8= LUkoZGlzcGxheUc2IjYpSSNiMUdGJEkjYjJHRiRJI2IzR0YkSSNiNEdGJEkjYjVHRiQvSSVheGVzR0YkSSdub3JtYWxHJSpwcm90ZWN0ZWRHL0koc2NhbGluZ0dGJEksY29uc3RyYWluZWRHRiQ= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Here we demonstrate the geometry of scalar multiplication in three space: QyQ+SSNjMUc2Ii1JJmFycm93R0YlNiVJI3YzR0YlL0kmd2lkdGhHRiUkIiIiISIiL0kmY29sb3JHRiVJJmdyZWVuR0YlRi4= QyQ+SSNjMkc2Ii1JJmFycm93R0YlNiUtSSItRzYkJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUdGJUkvVmVjdG9yQ2FsY3VsdXNHNiVGLEYsL0YuSShTdHVkZW50RzYkRixJKF9zeXNsaWJHRiU2Iy1JIipHRis2JCQiIzohIiJJI3YzR0YlL0kmd2lkdGhHRiUkIiIiRjsvSSZjb2xvckdGJUkkcmVkR0YlRjs= QyQ+SSNjM0c2Ii1JJmFycm93R0YlNiUtSSIqRzYkJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUdGJUkvVmVjdG9yQ2FsY3VsdXNHNiVGLEYsL0YuSShTdHVkZW50RzYkRixJKF9zeXNsaWJHRiU2JCQiIiYhIiJJI3YzR0YlL0kmd2lkdGhHRiUkIiIiRjgvSSZjb2xvckdGJUkneWVsbG93R0YlRjg= LUkoZGlzcGxheUc2IjYmPCVJI2MxR0YkSSNjMkdGJEkjYzNHRiQvSSVheGVzR0YkSSdub3JtYWxHJSpwcm90ZWN0ZWRHL0koc2NhbGluZ0dGJEksY29uc3RyYWluZWRHRiQvSSV2aWV3R0YkNyU7LUkiLUc2JEYtL0krbW9kdWxlbmFtZUdGJEkvVmVjdG9yQ2FsY3VsdXNHNiVGLUYtL0Y5SShTdHVkZW50RzYkRi1JKF9zeXNsaWJHRiQ2IyIiJUZBRjRGNA== The standard unit vectors in three space are displayed below: QyQ+SSJpRzYiLUkmYXJyb3dHRiU2JS1JJDwsPkdGJTYlIiIiIiIhRi0vSSZ3aWR0aEdGJSRGLCEiIi9JJmNvbG9yR0YlSSZncmVlbkdGJUYx QyQ+SSJqRzYiLUkmYXJyb3dHRiU2JS1JJDwsPkdGJTYlIiIhIiIiRiwvSSZ3aWR0aEdGJSRGLSEiIi9JJmNvbG9yR0YlSSRyZWRHRiVGMQ== QyQ+SSJrRzYiLUkmYXJyb3dHRiU2JS1JJDwsPkdGJTYlIiIhRiwiIiIvSSZ3aWR0aEdGJSRGLSEiIi9JJmNvbG9yR0YlSSVibHVlR0YlRjE= LUkoZGlzcGxheUc2IjYoSSJpR0YkSSJqR0YkSSJrR0YkL0klYXhlc0dGJEknbm9ybWFsRyUqcHJvdGVjdGVkRy9JKHNjYWxpbmdHRiRJLGNvbnN0cmFpbmVkR0YkL0kldmlld0dGJDclOy1JIi1HNiRGLC9JK21vZHVsZW5hbWVHRiRJL1ZlY3RvckNhbGN1bHVzRzYkRixJKF9zeXNsaWJHRiQ2IyIiI0Y9RjNGMw==

10. There are several ways to calculate the length of a vector with Maple V. Recall that V = <LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEiYkYnRi9GMkY5> then the length of V, ||V|| , is ||V|| = 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. Thus the most natural way to obtain the length of the vector , u defined above, is as follows: LUklc3FydEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLUkiK0c2JEYlL0krbW9kdWxlbmFtZUdGJ0kvVmVjdG9yQ2FsY3VsdXNHNiVGJUYlL0YtSShTdHVkZW50R0YkNiQqJCZJI3UyR0YnNiMiIiIiIiMqJCZGNTYjRjhGOA== Or if you want this result in floating point decimal to 15 digits: LUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiRJIiVHNiIiIzo= 11. For a vector in 3 space V = <a,b,c>, length of V is computed as ||V|| = 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. QyQtSSVzcXJ0RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMtSSIrRzYkRiYvSSttb2R1bGVuYW1lR0YoSS9WZWN0b3JDYWxjdWx1c0c2JUYmRiYvRi5JKFN0dWRlbnRHRiU2JC1GKzYkKiQmSSN1M0dGKDYjIiIiIiIjKiQmRjg2I0Y7RjsqJCZGODYjIiIkRjtGOg==LUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiNJIiVHNiI= 12. The built-in function to determine the length of a vector is the Norm function (2nd norm). There are several different norms for vectors -- the one for length being the second norm which is the default. LUklTm9ybUc2IjYjSSN1MkdGJA==

13. The vector operation dot product (section 10.3) is available in VectorCalculus. The dot product of two vectors u = < 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> and v = < 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> in the plane is defined to be 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. LUkwZGVsYXlEb3RQcm9kdWN0RzYkJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHNiRGJUkoX3N5c2xpYkdGKDYkSSN1MkdGKEkjdjJHRig= LUkwZGVsYXlEb3RQcm9kdWN0RzYkJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHNiRGJUkoX3N5c2xpYkdGKDYkSSN1MkdGKEYt 14. This allows us to calculate the length of a vector by yet another method. LUklc3FydEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLUkwZGVsYXlEb3RQcm9kdWN0RzYkRiUvSSttb2R1bGVuYW1lR0YnSSxUeXBlc2V0dGluZ0dGJDYkSSN1MkdGJ0Yw JSFH

16. One can use scalar multipllication to normalize a vector: (i.e find a vector of length one in same direction.) PkkiVUc2Ii1JIipHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YkSS9WZWN0b3JDYWxjdWx1c0c2JUYoRigvRipJKFN0dWRlbnRHNiRGKEkoX3N5c2xpYkdGJDYkSSN1MkdGJCokLUklc3FydEdGLzYjLUkwZGVsYXlEb3RQcm9kdWN0RzYkRigvRipJLFR5cGVzZXR0aW5nR0YvNiRGMkYyISIi LUklTm9ybUc2IjYjSSJVR0Yk JSFH

17. The angle between two vectors can be computed using the dot product by the formula 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. Let u = <LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1JI21vR0YkNi1RKiZ1bWludXMwO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOi8lKXN0cmV0Y2h5R0Y6LyUqc3ltbWV0cmljR0Y6LyUobGFyZ2VvcEdGOi8lLm1vdmFibGVsaW1pdHNHRjovJSdhY2NlbnRHRjovJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZJLUkjbW5HRiQ2JFEiMUYnRjVGNS1GMjYtUSIsRidGNUY4L0Y8USV0cnVlRidGPUY/RkFGQ0ZFL0ZIUSYwLjBlbUYnL0ZLUSwwLjMzMzMzMzNlbUYnLUZNNiRRIjNGJ0Y1RjU= > and v = < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW5HRiQ2JFEiMkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIixGJ0YvLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR1EldHJ1ZUYnLyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiRRIjRGJ0YvRi8=> be two vectors in the plane. 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 18. Calculate the dot product of the two vectors: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNictRiw2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIi5GJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUy1GLDYlUSJ2RidGNkY5LyUrZXhlY3V0YWJsZUdGREZARitGWUZARitGWUZA 19. Compute the length of each vector: 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1JJm1zcXJ0R0YkNiMtRiM2Jy1GLDYlUSJ1RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLkYnL0Y9USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRy8lKXN0cmV0Y2h5R0ZHLyUqc3ltbWV0cmljR0ZHLyUobGFyZ2VvcEdGRy8lLm1vdmFibGVsaW1pdHNHRkcvJSdhY2NlbnRHRkcvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZWRjYvJStleGVjdXRhYmxlR0ZHRkNGWUZDRitGWUZD 20. Compute the cosine of the angle between the two vectors 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 21. Compute the angle between the two vectors (answer is in radians): LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNictRiw2JVEmZXZhbGZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUTAmQXBwbHlGdW5jdGlvbjtGJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUy1JKG1mZW5jZWRHRiQ2JC1GIzYnRistRiM2Jy1GLDYlUSdhcmNjb3NGJy9GN0ZERkBGPC1GVzYkLUYjNiUtRiw2JVEiQ0YnRjZGOS8lK2V4ZWN1dGFibGVHRkRGQEZARmJvRkBGK0Zib0ZARkBGYm9GQEYrRmJvRkBGK0Zib0ZA 22. In degrees this would be: 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

1. Define vectors C as < 4, -3> and B as <5, 2>. Then calculate the a) C + B b) 3C + 2B c) length of C d) length of 3C e) a vector in the same direction as C but with unit length. f) the dot product of C and B g) the angle between C and B in both radians and degrees