Project a point onto an intersection line

Relative to the origin OO, the point AA has position vector OA→=(0,1,0)\overrightarrow{OA}=(0,1,0) and AA lies on the plane Π1\Pi_1 with equation defined by r⋅(1,3,2)=3\mathbf r\cdot(1,3,2)=3. Another plane Π2\Pi_2 has equation y=xy=x. The planes Π1,Π2\Pi_1,\Pi_2 intersect at line ll.

(i) Find a vector equation of the line ll. mark

(ii) Show that the cosine of the acute angle between the planes Π1,Π2\Pi_1,\Pi_2 is 77\frac{\sqrt7}{7}. marks

(iii) Find the position vector of the foot of perpendicular, OF→\overrightarrow{OF}, from point AA to the line ll. Hence, find the exact length of projection of AFAF onto the plane Π2\Pi_2. marks

(iv) Another plane Π3\Pi_3 has equation px+qy=1px+qy=1, where p,qp,q are real constants. Find the condition which p,qp,q must satisfy such that ll intersects Π3\Pi_3 at exactly one point. marks

[2010/NJC/Prelim/I/9]

Total marks: 10

Solution

(i)

Line of intersection

Π1:x+3y+2z=3\Pi_1:x+3y+2z=3 and Π2:y=x\Pi_2:y=x meet in ll. Find a vector equation of ll.

Explore this idea

Hint

  1. Direction =(1,3,2)×(−1,1,0)=(1,3,2)\times(-1,1,0).

  2. Find a point with z=0z=0.

Answer · (i)

r=(3/4,3/4,0)+t(1,1,−2)r=(3/4,3/4,0)+t(1,1,-2).

Explore the idea

Two planes and the line l

Switch parts of the scene on one at a time and read how each is found. Drag to turn the picture.

Open the interactive view to explore this geometry in 3D.

(ii)

Angle between planes

Show that the cosine of the acute angle between Π1\Pi_1 and Π2\Pi_2 is 77\frac{\sqrt7}7.

Hint

  1. Angle between the normals (1,3,2)(1,3,2) and (−1,1,0)(-1,1,0).

Answer · (ii)

cos⁡θ=7/7\cos\theta=\sqrt7/7.

(iii) foot

Perpendicular foot on a line

Find the position vector of the foot FF of the perpendicular from A(0,1,0)A(0,1,0) to ll.

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Hint

  1. OF→=(34,34,0)+λ(1,1,−2)\overrightarrow{OF}=(\frac34,\frac34,0)+\lambda(1,1,-2).

  2. AF→⋅d=0\overrightarrow{AF}\cdot\mathbf d=0.

Answer · (iii) foot

F=(2/3,2/3,1/6)F=(2/3,2/3,1/6).

Explore the idea

Follow the foot F on the intersection line

Move A to explore perpendicular feet on the fixed line l. Free movement may take A off Π₁; Reset restores A = (0, 1, 0) and F = (2/3, 2/3, 1/6) before the next part's plane-projection calculation.

Open the interactive view to explore this geometry in 3D.

(iii) projection

Projection onto a plane

Hence find the exact length of the projection of AFAF onto Π2\Pi_2.

Hint

  1. ∣AF∣=216|AF|=\frac{\sqrt{21}}6.

  2. Projection =∣AF∣cos⁡θ=|AF|\cos\theta.

Answer · (iii) projection

3/6\sqrt3/6.

(iv)

One point of intersection

Π3:px+qy=1\Pi_3:px+qy=1. Find the condition on p,qp,q such that ll meets Π3\Pi_3 at exactly one point.

Hint

  1. A line meets a plane at exactly one point unless it is parallel to it: d⋅n≠0\mathbf d\cdot\mathbf n\ne0.

Answer · (iv)

p+q≠0p+q\ne0.