Subjective Type

How do you find two unit vectors orthogonal ot both $$i-j+k$$ and $$4j+4k$$?

Solution

for vectors $$\vec { a } $$ and $$\vec { b } $$ their vector
(cross ) product ie $$\vec { a } \times \vec { b } $$ is orthogonal to both $$\vec { a } $$ and $$\vec { b } $$
the desired unit vectors can be obtained by
$$\pm \cfrac { \vec { a } \times \vec { b } }{ \left\| \vec { a } \times \vec { b } \right\| } $$
$$\vec { a } =i-j+k=(1,-1,1);\vec { b } =4j+4k=4(0,1,1)$$
$$\vec { a } \times \vec { b } =4\begin{vmatrix} i & j & k \\ 1 & -1 & 1 \\ 0 & 1 & 1 \end{vmatrix}=4(-2,-1,1)$$
$$\left\| \vec { a } \times \vec { b } \right\| =4\sqrt { { \left( -2 \right) }^{ 2 }+{ \left( -1 \right) }^{ 2 }+{ \left( 1 \right) }^{ 2 } } =4\sqrt { 6 } \quad $$
desired vectors are $$\cfrac { \pm 4(-2,-1,1) }{ 4\sqrt { 6 } } $$ or $$\cfrac { 1 }{ \pm \sqrt { 6 } } (-2,-1,1)$$

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