a) To find the dot product of the vectors u and v, we can use the Dot function in Mathematica. The dot product is calculated as follows:
u.v = Dot[u, v]
b) To find the angle between vectors u and v, we can use the ArcCos function in Mathematica. The angle is calculated as follows:
angle = ArcCos[(u.v)/(Norm[u]*Norm[v])]
c) We can graph both vectors u and v on the same system using the ListVectorPlot3D function in Mathematica. This will display the vectors in a 3D coordinate system.
ListVectorPlot3D[{u, v}]
d) To graph vectors u, v, and u * v on the same set of axes with different colors, we can use the Graphics3D function in Mathematica. We can assign a different color to u * v using the Directive function.
Graphics3D[{Arrow[{{0, 0, 0}, u}], Arrow[{{0, 0, 0}, v}],
Directive[Red], Arrow[{{0, 0, 0}, u*v}]}]
e) To rotate the graph from part d and show two different views of the cross product, we can use the ViewPoint option in the Graphics3D function. By specifying different viewpoints, we can obtain different perspectives of the graph.
Graphics3D[{Arrow[{{0, 0, 0}, u}], Arrow[{{0, 0, 0}, v}],
Directive[Red], Arrow[{{0, 0, 0}, u*v}]},
ViewPoint -> {1, -1, 1}]
Graphics3D[{Arrow[{{0, 0, 0}, u}], Arrow[{{0, 0, 0}, v}],
Directive[Red], Arrow[{{0, 0, 0}, u*v}]},
ViewPoint -> {1, 1, 1}]
f) To find the normal vector to vector u, we can use the Cross function in Mathematica. The normal vector is calculated as follows:
normal = Cross[u]
The function Cross[u] computes the cross product of u with the standard basis vectors. The resulting vector represents the direction perpendicular to the plane spanned by u.
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