Plane intersections and orthogonal directions

Planes Π1,Π2\Pi_1,\Pi_2 are r⋅(2,4,1)=10\mathbf r\cdot(2,4,1)=10 and r⋅(1,3,1)=8\mathbf r\cdot(1,3,1)=8.

(i) Find their acute angle. marks

(ii) Obtain a vector equation of intersection line l1l_1. marks

Another line l2l_2 is (x−2)/6=(7−z)/3(x-2)/6=(7-z)/3, y=my=m, m real.

(iii) If Π1\Pi_1 and l2l_2 intersect at (6,m,5)(6,m,5), find m. marks

(iv) Show l1,l2l_1,l_2 are perpendicular for all m. marks

[2010/DHS/Prelim/II/4]

Total marks: 11

Solution

(i)

Angle between planes

Find the acute angle between Π1:r⋅(2,4,1)=10\Pi_1:\mathbf r\cdot(2,4,1)=10 and Π2:r⋅(1,3,1)=8\Pi_2:\mathbf r\cdot(1,3,1)=8.

Explore this idea

Hint

  1. The angle between planes is the acute angle between their normals.

Answer · (i)

cos⁡−1(15/231)=9.3∘\cos^{-1}(15/\sqrt{231})=9.3^\circ.

Explore the idea

Two planes meeting at a small angle

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)

Line of intersection

Find a vector equation of the line l1l_1 where Π1\Pi_1 and Π2\Pi_2 meet.

Explore this idea

Hint

  1. Direction =n1×n2=\mathbf n_1\times\mathbf n_2.

  2. Find one common point by setting z=0z=0.

Answer · (ii)

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

Explore the idea

The line l₁ common to both planes

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.

(iii)

Line meeting a plane

l2:x−26=7−z3l_2:\frac{x-2}6=\frac{7-z}3, y=my=m meets Π1\Pi_1 at (6,m,5)(6,m,5). Find mm.

Hint

  1. The point lies on Π1\Pi_1: substitute into 2x+4y+z=102x+4y+z=10.

Answer · (iii)

m=−7/4m=-7/4.

(iv)

Perpendicular directions

Show that l1l_1 and l2l_2 are perpendicular for all mm.

Explore this idea

Hint

  1. l2l_2: r=(2,m,7)+β(2,0,−1)\mathbf r=(2,m,7)+\beta(2,0,-1).

Answer · (iv)

(1,−1,2)⋅(6,0,−3)=0(1,-1,2)\cdot(6,0,-3)=0: directions are orthogonal for every mm.

The lines intersect only for m=−2/5m=-2/5; otherwise they are skew, not intersecting perpendicular lines.

Explore the idea

l₁ and l₂ (m = −7/4)

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.