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Calculus IIChapter 6: 3-Dimensional SpaceSection 6-2

Equations of Lines — Question 2

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Question 2

Determine whether the lines L1:r→=⟨1,0,2⟩+s⟨2,−1,3⟩,L2:r→=⟨3,−1,1⟩+t⟨1,1,2⟩L_1:\ \vec r=\left\langle 1,0,2\right\rangle+s\left\langle 2,-1,3\right\rangle,\qquad L_2:\ \vec r=\left\langle 3,-1,1\right\rangle+t\left\langle 1,1,2\right\rangle intersect, are parallel, or are skew. Justify the classification.

Original worksheet page 1: question and worked solution for 6-2-002
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Question 2 – Solution

Strategy Equate coordinates. An intersection requires one pair (s,t)(s,t) satisfying all three equations.

See the diagram in the original worksheet below.

Coordinate equations The first two equations are 1+2s=3+t1+2s=3+t and −s=−1+t-s=-1+t. The second gives t=1−st=1-s; substitution into the first gives s=1s=1, hence t=0t=0.

Consistency check The third coordinates would be 2+3(1)=52+3(1)=5 and 1+2(0)=11+2(0)=1, so they disagree. The directions are not scalar multiples. Therefore the lines are skew\boxed{\text{skew}}.

Key idea Solving only two coordinate equations can produce a false intersection; the remaining coordinate is essential.

Original worksheet page 2: question and worked solution for 6-2-002

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