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प्रश्न
Find the Cartesian equations of the line passing through the point A(1, 1, 2) and perpendicular to the vectors `barb = hati + 2hatj + hatk and barc = 3hati + 2hatj - hatk`.
उत्तर
Let the required line have direction ratios p, q, r
It is perpendicular to the vectors `barb = hati + 2hatj + hatk and barc = 3hati + 2hatj - hatk`
∴ It is perpendicular to lines whose direction ratios are 1, 2, 1 and 3, 2, – 1
∴ p + 2q + r = 0, 3p + 2q – r = 0
∴ `p/|(2, 1),(2, -1)| = q/(-|(1, 1),(3, -1)|) = r/|(1, 2),(3, 2)|`
∴ `p/(-4) = q/(4) = r/(-4)`
∴ `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)/(-1)`,
i.e. x – 1 = y – 1 = z – 2
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