Two lines in 3D Cartesian space

Geometry Level 2

If the two lines x − 1 k = y + 1 2 = z , x + 2 − 3 = 1 − y = z + 2 k \begin{aligned} \frac {x-1 }{k } &=\frac { y+1 }{2 } =z,\\ \\ \\ \frac {x+2 }{-3 } & =1-y =\frac{z+2}{k} \end{aligned} are perpendicular to each other, then what is the value of k ? k?


The answer is -1.

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Lawahar Qasid by Lawahar Qasid

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1 solution

Andrew Ellinor
Oct 16, 2015

If two lines are perpendicular, then the vectors corresponding to their direction are perpendicular, meaning their dot product is 0.

The first line's direction vector is ⟨ k , 2 , 1 ⟩ \langle k, 2, 1\rangle and the second line's direction vector is ⟨ − 3 , − 1 , k ⟩ \langle -3, -1, k\rangle .

⟨ k , 2 , 1 ⟩ ⋅ ⟨ − 3 , − 1 , k ⟩ = − 3 k − 2 + k = 0 ⟶ k = − 1 \langle k, 2, 1\rangle \cdot \langle -3, -1, k\rangle = -3k - 2 + k = 0 \longrightarrow k = -1

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But when i text -1 it replied as " your ans should be an integer"

Lahkadhirsinh Gohil - 5 years, 5 months ago

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