If [tex]l_1, m_1, n_1[/tex] and [tex]l_2, m_2, n_2[/tex] are the direction cosines of two lines, show that the direction cosines of the line perpendicular to them are proportional to [tex]m_1 n_2 - m_2 n_1, n_1 l_2 - n_2 l_1, l_1 m_2 - l_2 m_1[/tex].

The direction cosines of a line that is perpendicular to two lines with specified direction cosines can be found using the cross product of the original lines' direction cosines. This resulting vector is proportional to the original vectors, hence proving that the direction cosines of the perpendicular line are proportional to the cross product.

Explanation:

In the field of mathematics, particularly in vector calculus, the direction cosines of a vector are the cosines of the angles it makes with the coordinate axes. In this case, let's consider two lines with direction cosines l1,m1,n1 and l2,m2,n2. The direction cosines of a line are associated with the angles the line makes with the principal axes.

If we want to find the direction cosines of a line that is perpendicular to both lines, we can use the formula for the cross product of two vectors. The new line would be proportional to the cross product of the original lines, given as:

(m1n2−m2n1, n1l2−n2l1, l1m2−l2m1).

This is because the cross product of two vectors gives a vector that is perpendicular to both original vectors, and the coefficients of the result are proportional to the sine of the angle between the vectors. Thus, in this context, 'proportional' refers to the consistent ratio of the dimensions of the resulting vector in relation to the original vectors.

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