Planes containing a prescribed line

The equations of the line l1l_1 and the plane p1p_1 are respectively, r=(1,0,0)+μ(−1,1,1)\mathbf r=(1,0,0)+\mu(-1,1,1) and r=s(1,1,1)+t(2,−1,1)\mathbf r=s(1,1,1)+t(2,-1,1), where μ,s,t∈R\mu,s,t\in\mathbb R.

(i) Find the acute angle between l1,p1l_1,p_1. marks

(ii) A second plane p2p_2 has equation r⋅(α,1,β)=1\mathbf r\cdot(\alpha,1,\beta)=1. Given that the two planes p1,p2p_1,p_2 intersect at the line l2:4x−15−2=y,z=52l_2:\frac{4x-15}{-2}=y,z=\frac52, find the values of α,β\alpha,\beta. marks

(iii) The plane p3p_3 with equation 2x+by+z=12x+by+z=1 is parallel to l2l_2. Find the value of bb. Hence find the distance between l2,p3l_2,p_3. marks

[2010/CJC/Prelim/II/2]

Total marks: 10

Solution

(i)

Angle between a line and a plane

l1:r=(1,0,0)+μ(−1,1,1)l_1:\mathbf r=(1,0,0)+\mu(-1,1,1) and p1:r=s(1,1,1)+t(2,−1,1)p_1:\mathbf r=s(1,1,1)+t(2,-1,1). Find the acute angle between l1l_1 and p1p_1.

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Hint

  1. Normal of p1=(1,1,1)×(2,−1,1)p_1=(1,1,1)\times(2,-1,1).

  2. Angle with the normal, then 90∘−90^\circ- that.

Answer · (i)

sin⁡−1(4/42)=38.1∘\sin^{-1}(4/\sqrt{42})=38.1^\circ.

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A line and a plane

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(ii)

Plane containing a line

p2:r⋅(α,1,β)=1p_2:\mathbf r\cdot(\alpha,1,\beta)=1 meets p1p_1 in l2:4x−15−2=y, z=52l_2:\frac{4x-15}{-2}=y,\ z=\frac52. Find α\alpha and β\beta.

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Hint

  1. l2l_2 lies in p2p_2: its direction is perpendicular to the normal and its point satisfies the equation.

Answer · (ii)

α=2,β=−13/5\alpha=2,\beta=-13/5.

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The line l₂ in both planes

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(iii)

Parallel plane and distance

p3:2x+by+z=1p_3:2x+by+z=1 is parallel to l2l_2. Find bb and the distance between l2l_2 and p3p_3.

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Hint

  1. Parallel: (−1,2,0)⋅(2,b,1)=0(-1,2,0)\cdot(2,b,1)=0.

  2. Distance from a point of l2l_2 to p3p_3.

Answer · (iii)

b=1b=1; distance 36/23\sqrt6/2.

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A line parallel to a plane

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