Skew lines and a perpendicular foot · source reference

Question

Lines $l_1,l_2$ are $\mathbf r=(1,0,1)+\lambda(1,1,-2)$ and $\mathbf r=(1,2,-1)+\mu(7,5,1)$ respectively.

(i) Show they are skew lines. [3]

(ii) Find the position vector of the foot of perpendicular from A$(1,2,-1)$ to $l_1$. [3]

[modified 2009/NYJC/Prelim/I/3]

Source working

(i) The direction vectors $(1,1,-2)$ and $(7,5,1)$ are not parallel, so the lines may intersect. Equating line coordinates:

$$\lambda-7\mu=0\quad(1),\quad\lambda-5\mu=2\quad(2),\quad2\lambda+\mu=2\quad(3).$$

Solving (1),(2) using GC gives $\mu=1,\lambda=7$. Substitute into unused third equation: LHS $=2(7)+1=15\ne2=$ RHS. The source concludes: “Since there is no unique solution, the lines are skew lines.”

Qualification: The computed inconsistency shows there is no solution, rather than merely “no unique solution”. Nonparallel lines that do not intersect are skew.

(ii) Let N be the foot. N on $l_1$:

$$\overrightarrow{ON}=(1,0,1)+\lambda(1,1,-2),\quad\lambda\in\mathbb R.$$

$AN\perp l_1\Rightarrow\overrightarrow{AN}\cdot(1,1,-2)=0$:

$$[(1,0,1)+\lambda(1,1,-2)-(1,2,-1)]\cdot(1,1,-2)=0,$$ $$[(0,-2,2)+\lambda(1,1,-2)]\cdot(1,1,-2)=0,$$ $$(-2-4)+\lambda(1+1+4)=0\Rightarrow\lambda=1.$$

Thus $\overrightarrow{ON}=(2,1,-1)$.

Teacher notes

In $\mathbf r=(1,0,1)+\lambda(1,1,-2)$, the first vector is the position vector of a known point on the line; the second is its direction vector; $\lambda\in\mathbb R$.

Solve for μ, λ using two coordinate equations and substitute into the third to check satisfaction.

For the perpendicular foot, use two conditions: N lies on $l_1$ so satisfies the line equation; $AN\perp l_1$, so $\overrightarrow{AN}\cdot\mathbf d=0$, with d its direction vector.

Native diagrams

The source shows a straight $l_1$ through N, a perpendicular AN with A$(1,2,-1)$ above, a right-angle mark at N and direction arrow $(1,1,-2)$ along the line.

Printed answer key

(ii) $(2,1,-1)$.

Provenance

2025 TJC H2 Math (9758) JC2 MSM: VECTORS. Question document pages 1; solution document pages 4. Original page numbers are retained as text metadata.