Answer :

__Given:__ line passes through (1, 2, -4) and is perpendicular to each of the lines and

__To find:__ equation of line in Vector and Cartesian form

__Formula Used:__ Equation of a line is

Vector form:

Cartesian form:

where is a point on the line and is a vector parallel to the line.

If 2 lines of direction ratios a_{1}:a_{2}:a_{3} and b_{1}:b_{2}:b_{3} are perpendicular, then a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3} = 0

__Explanation:__

Here,

Let the direction ratios of the line be b_{1}:b_{2}:b_{3}

Direction ratios of other two lines are 8 : -16 : 7 and 3 : 8 : -5

Since the other two line are perpendicular to the given line, we have

8b_{1} – 16b_{2} + 7b_{3} = 0

3b_{1} + 8b_{2} – 5b_{3} = 0

Solving,

Therefore,

Vector form of the line is:

Cartesian form of the line is:

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