Find the Cartesian equations of the line passing through the point A(1, 1, 2) and perpendicular to the vectors bijkandcijkb¯=i^+2j^+k^andc¯=3i^+2j^-k^ - Mathematics and Statistics

Sum

Find the Cartesian equations of the line passing through the point A(1, 1, 2) and perpendicular to the vectors bar"b" = hat"i" + 2hat"j" + hat"k" and bar"c" = 3hat"i" + 2hat"j" - hat"k"

Solution

Let the required line have direction ratios p, q, r.

It is perpendicular to the vector bar"b" = hat"i" + 2hat"j" + hat"k" and bar"c" = 3hat"i" + 2hat"j" - hat"k".

∴ it is perpendicular to lines whose direction ratios are 1, 2, 1 and 3, 2, – 1.

∴ p + 2q + r = 0, 3 + 2q – r = 0

∴ p/|(2, 1),(2, -1)| = q/|(1, 1),(-1, 3)| = r/|(1, 2),(3, 2)|

∴ p/(-4) = q/(4) = r/(-1)

∴ p/(-1) = q/(1) = r/(-1)

∴ the required line has direction ratios –1, 1, –1.

The cartesian equations of the line passing through (x1, y1, z1) and having direction ratios a, b, c are (x = x_1)/a = (y - y_1)/b = (z - z_1)/c

∴ the cartesian equation of the line passing through the point (1, 1, 2) and having directions ratios –1, 1, – 1 are (x - 1)/(-1) = (y - 1)/(1) = (z - 2)/(-2).

Concept: Vector and Cartesian Equations of a Line
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Balbharati Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board
Chapter 6 Line and Plane
Miscellaneous Exercise 6 A | Q 8 | Page 208