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Xelga [282]
3 years ago
13

Find two unit vectors orthogonal to both given vectors. i j k, 4i k

Mathematics
1 answer:
Maurinko [17]3 years ago
6 0
The cross product of two vectors gives a third vector \mathbf v that is orthogonal to the first two.

\mathbf v=(\vec i+\vec j+\vec k)\times(4\,\vec i+\vec k)=\begin{vmatrix}\vec i&\vec j&\vec k\\1&1&1\\4&0&1\end{vmatrix}=\vec i+3\,\vec j-4\,\vec k

Normalize this vector by dividing it by its norm:

\dfrac{\mathbf v}{\|\mathbf v\|}=\dfrac{\vec i+3\,\vec j-4\,\vec k}{\sqrt{1^2+3^2+(-4)^2}}=\dfrac1{\sqrt{26}}(\vec i+3\,\vec j-4\vec k)

To get another vector orthogonal to the first two, you can just change the sign and use -\mathbf v.
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