# Find the Cartesian and vector equations of the line passing through the point (1, 2, -4) and perpendicular to each of the lines and

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 a1:a2:a3 and b1:b2:b3 are perpendicular, then a1b1+a2b2+a3b3 = 0

Explanation:

Here,

Let the direction ratios of the line be b1:b2:b3

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

8b1 – 16b2 + 7b3 = 0

3b1 + 8b2 – 5b3 = 0

Solving,

Therefore,

Vector form of the line is:

Cartesian form of the line is:

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