Find the value of k for which the lines (1 − x)/2 = (2y − 3)/k = (z + 2)/3 and (x − 4)/1 = (5 − y)/2 = (2z + 1)/4 are perpendicular to each other.
How to lay out the answer. Follow the frame, then write it in your own words.
1 · Write what is given and what is to be found. Given: two lines in Cartesian form that are perpendicular. To find: k.
2 · Draw the figure or sketch. Sketch the two lines L₁ and L₂, with their directions b⃗₁ and b⃗₂ marked at a right angle. Label both lines.
3 · Write the rule. Lines with DRs a₁, b₁, c₁ and a₂, b₂, c₂ are perpendicular when a₁a₂ + b₁b₂ + c₁c₂ = 0. First make the coefficients of x, y and z equal to +1.
L₁: (x − 1)/(−2) = (y − 3/2)/(k/2) = (z + 2)/3, so its DRs are −2, k/2, 3
(5 − y)/2 = (y − 5)/(−2) and (2z + 1)/4 = (z + 1/2)/2
L₂: (x − 4)/1 = (y − 5)/(−2) = (z + 1/2)/2, so its DRs are 1, −2, 2
5 · Use the perpendicular condition.
(−2)(1) + (k/2)(−2) + (3)(2) = 0
−2 − k + 6 = 0
k = 4
6 · Check and box the answer. With k = 4 the DRs are −2, 2, 3 and 1, −2, 2: (−2)(1) + (2)(−2) + (3)(2) = −2 − 4 + 6 = 0. So k = 4.
Scoring tip: Most marks are lost by reading 2 instead of −2 from (1 − x)/2, or k instead of k/2 from (2y − 3)/k. Rewrite both lines in standard form first.
