Find two unit vectors orthogonal to both j−k and i+j.

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intacte87 · Accepted Answer

Step 1Consider the vectors,j−k&amp;nbsp;and&amp;nbsp;i+jThe objective is to find two vectros orthogonal to both the vectors.From the vectors,Let,a=j−k=⟨0,&amp;nbsp;1,&amp;nbsp;−1⟩b=i+j=⟨1,&amp;nbsp;1,&amp;nbsp;0⟩Definition:Let a and b two vectors, the cross product&amp;nbsp;(a×b)&amp;nbsp;of a and b is orthogonal to both the vector a and bCalculate the cross product.a×b=|ijk01−1110|=(0−(−1)0i−(0−(−1))j+(0−1)k&amp;nbsp;=i−j−kThe orthogonal vector for the given two wectors is&amp;nbsp;i−j−kAnd∥a×b∥=(1)2+(−1)2+(−1)2=1+1+1=3Step 2Find two vectors orthogonal to both the vectors as follows:Then, it is possible to obtain the desired unit vector by,1)&amp;nbsp;±(a×b∥a×b∥)Substitute&amp;nbsp;a×b=i−j−k&amp;nbsp;and&amp;nbsp;∥a×b∥=3&amp;nbsp;in equation (1)±(a×b∥a×b∥)=±(i−j−k3)=±(13i−13j−13k)=+(13i−13j−13k)&amp;nbsp;or&amp;nbsp;−(13i−13j−13k)=13i−13j−13k&amp;nbsp;or&amp;nbsp;−13i+13j+13k=⟨13,&amp;nbsp;−13,&amp;nbsp;−13⟩&amp;nbsp;or&amp;nbsp;⟨−13,&amp;nbsp;13,&amp;nbsp;13⟩Consequently, the necessary vectors that are orthogonal to both vectors are,⟨13,&amp;nbsp;−13,&amp;nbsp;−13⟩&amp;nbsp;or&amp;nbsp;⟨−13,&amp;nbsp;13,&amp;nbsp;13⟩&amp;nbsp;

Beverly Smith · Accepted Answer

Step 1&amp;nbsp;Ut&amp;nbsp;a→=j−k&amp;nbsp;and&amp;nbsp;b→=i+j&amp;nbsp;=⟨0,&amp;nbsp;1,&amp;nbsp;−1⟩&amp;nbsp;=⟨1,&amp;nbsp;1,&amp;nbsp;0⟩&amp;nbsp;The cross product&amp;nbsp;a→×b→&amp;nbsp;is ostrogonal to both&amp;nbsp;a→&amp;nbsp;and&amp;nbsp;b→&amp;nbsp;a→×b→=|ijk01−1110|&amp;nbsp;=i(0−(−1))−j(0−(−1))+k(0−1)&amp;nbsp;=i−j−k&amp;nbsp;A vector with a unit length&amp;nbsp;The unit vector is shown there:i−j−k12+(−1)2+(−1)2=i−j−k3&amp;nbsp;=13i−13j−13k&amp;nbsp;Step 2&amp;nbsp;The second unot vector osthogonal ot both&amp;nbsp;a→&amp;nbsp;and&amp;nbsp;b→&amp;nbsp;would be negative of the previous vector&amp;nbsp;=−13i+13j+13k&amp;nbsp;Therefore, the two unit vectors osthogonal to both&amp;nbsp;j−k&amp;nbsp;and&amp;nbsp;i+j&amp;nbsp;are&amp;nbsp;i3−j3−k3&amp;nbsp;and&amp;nbsp;−i3+j3+k3

nick1337 · Accepted Answer

Step 1 j−k=&amp;lt;&amp;lt;0, 1, −1&amp;gt;&amp;gt; i+j=&amp;lt;&amp;lt;1, 1, 0&amp;gt;&amp;gt; The cross product will produce a vector orthogonal to both,  &amp;lt;&amp;lt;0, 1, −1&amp;gt;&amp;gt;×&amp;lt;&amp;lt;1, 1, 0&amp;gt;&amp;gt;=&amp;lt;&amp;lt;1(0)−(−1)(1)−(0)(0),(0)(1)−(1)(1)&amp;gt;&amp;gt; v=&amp;lt;&amp;lt;1, −1, −1&amp;gt;&amp;gt; Step 2 Find the length of v |v|=12+(−1)2+(−1)2=3 Divide the components of v by this to make it have unit length (normalize it) v|v|=&amp;lt;&amp;lt;13, −13, −13&amp;gt;&amp;gt;=&amp;lt;&amp;lt;33, −33, −33&amp;gt;&amp;gt;=33i−33j−33k To get the other orthogonal vector we can just negate this to become: −33i+33j+33k

Find two unit vectors orthogonal to both j - k

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