Find the vector and Cartesian equations of the line through (1, 2, −4) parallel to the line (x − 3)/2 = (4 − y)/5 = (2z + 1)/6.
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: the point (1, 2, −4) and a line in Cartesian form. To find: both equations of the parallel line.
2 · Draw the figure or sketch. Sketch two parallel lines and mark the point (1, 2, −4) on the new one.
3 · Write the rule. Put the given line in standard form; parallel lines have the same (or proportional) DRs.
4 · Put the given line in standard form.
(4 − y)/5 = (y − 4)/(−5)
(2z + 1)/6 = (z + 1/2)/3
Standard form: (x − 3)/2 = (y − 4)/(−5) = (z + 1/2)/3, so the DRs are 2, −5, 3
5 · Write both equations.
Cartesian: (x − 1)/2 = (y − 2)/(−5) = (z + 4)/3
Vector: r⃗ = (î + 2ĵ − 4k̂) + λ(2î − 5ĵ + 3k̂)
6 · Check and box the answer. At λ = 0 the vector form gives (1, 2, −4), the given point. Box both equations.
Scoring tip: Most marks are lost by reading 5 instead of −5 from (4 − y)/5. Rewrite the line first.
