Mathematics

Mathematics

Problem

Determine the unit vector perpendicular to a=3i−j+2k\mathbf{a}=3\mathbf{i}-\mathbf{j}+2\mathbf{k} and b=−i+4j−k\mathbf{b}=-\mathbf{i}+4\mathbf{j}-\mathbf{k} that has a positive zz-component. a) 1171 (7i−j−11k)\dfrac{1}{\sqrt{171}}\,( 7\mathbf{i}-\mathbf{j}-11\mathbf{k} ) b) 1171 (−7i−j+11k)\dfrac{1}{\sqrt{171}}\,( -7\mathbf{i}-\mathbf{j}+11\mathbf{k} ) c) 1171 (−7i+j+11k)\dfrac{1}{\sqrt{171}}\,( -7\mathbf{i}+\mathbf{j}+11\mathbf{k} ) d) 1171 (−7i+j+11k)\dfrac{1}{171}\,( -7\mathbf{i}+\mathbf{j}+11\mathbf{k} )
Intuitive explanation

The cross product a×b\mathbf{a} \times \mathbf{b} gives you a vector perpendicular to both a\mathbf{a} and b\mathbf{b} simultaneously—that's the key insight. Once you compute it, check the sign of the zz-component; if it's negative, flip the direction. Then normalize by dividing by the magnitude to get your unit vector.

Intuitive explanation

The cross product a×b\mathbf{a} \times \mathbf{b} gives you a vector perpendicular to both a\mathbf{a} and b\mathbf{b} simultaneously—that's the key insight. Once you compute it, check the sign of the zz-component; if it's negative, flip the direction. Then normalize by dividing by the magnitude to get your unit vector.

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