Question

5. What are the two unit vectors perpendicular toand which vector is in the direction of the cross product l il × .?

Answer 1

The first vector is $5\vec{i}+\vec{j}$ and the second vector is $-\vec{i}-\vec{j}$ .

The unit vector having the same direction as the first vector is $5/\sqrt{25+1}\vec{i}+1/\sqrt{25+1}\vec{j}=5/\sqrt{26}\vec{i}+1/\sqrt{26}\vec{j}$ .

The unit vector perpendicular to this vector is $-1/\sqrt{26}\vec{i}+5/\sqrt{26}\vec{j}$ .

The unit vector having the same direction as the second vector is $-1/\sqrt{1^{2}+1^{2}}\vec{i}-1\sqrt{1^{2}+1^{2}}\vec{j}=-1/\sqrt{2}\vec{i}-1/\sqrt{2}\vec{j}$ .

The unit vector perpendicular to this vector is $1/\sqrt{2}\vec{i}-1/\sqrt{2}\vec{j}$ .

The cross product of these 2 vectors is $(-\vec{i}-\vec{j})*(5\vec{i}+\vec{j})=-\vec{i}*\vec{j}-5\vec{j}*\vec{i}=-\vec{k}+5\vec{k}=4\vec{k}$ .

Therefore the z-axis or $\vec{k}$ is in the direction of the cross product of these 2 vectors.