Vector calculus in plane and space: The Cross Product
The standard cross product
The cross product in coordinates
Let #\vec{e}_1#, #\vec{e}_2#, #\vec{e}_3# be a right-hand-oriented basis of the space and write the coordinates relative to the basis. The cross product of the vectors is then #\vec{v}=\rv{v_1, v_2, v_3}# and #\vec{w} = \rv{w_1, w_2, w_3}#, where the coordinates relative to the basis are given by \[\vec{v}\times\vec{w} = \rv{v_2\cdot w_3-v_3\cdot w_2, v_3\cdot w_1-v_1\cdot w_3, v_1\cdot w_2-v_2\cdot w_1}\]
Because the standard basis of #\mathbb{R}^3# is orthonormal, this formula particularly applies to the cross product in #\mathbb{R}^3# with respect to said basis. In order to distinguish this cross product from cross products expressed in coordinates given by other basic choices, we give it a special name:
Standard cross product
The standard cross product of two vectors #\vec{v}= \rv{v_1, v_2,v_3}# and #\vec{w}= \rv{w_1, w_2,w_3}# in #\mathbb{R}^3#, is the cross product expressed in terms of coordinates relative to the standard basis. It is given by the formula above.
A normal vector of the plane #V# with parametric representation #\vec{x} = \rv{1,2,3}+ \lambda\cdot\rv{1,2,1}+\mu \cdot\rv{3,1,0}# is \[\rv{1,2,1}\times \rv{3,1,0}= \rv{-1,3,-5}\tiny.\] An equation of the plane is then #-x_1 + 3x_2 -5x_3 = d# for one or other #d#. If we enter #\rv{1,2,3}#, we find that #d=-10#. An equation is #-x_1 +3x_2 -5x_3 = -10#.
Determine a normal vector of #V#.
Give your answer in the form #\rv{a_1,a_2,a_3}# wherein #a_1#, #a_2#, #a_3# are integers.
A normal vector of #V# is a vector that is not the zero vector and is perpendicular to the two directional vectors #\rv{-5,-5,-3}# and #\rv{2,-2,0}#. The cross product of these two vectors is one such vector. According to the cross product in coordinates the following applies:
\[\begin{array}{rcl}\rv{-5,-5,-3}\times\rv{2,-2,0}&=&\rv{-5\cdot 0+3\cdot -2, -3\cdot 2+5\cdot 0, -5\cdot -2+5\cdot 2}\\&=&\rv{-6,-6,20}\end{array}\]
Therefore, #\rv{-6,-6,20}# is a normal vector.
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