📝 Unit 9: Intersections

Line–Line · Line–Plane · Plane–Plane · Three planes
🔄 Not Graded
Purpose: Self-check on systems of linear equations from lines and planes.
Score: 0 / 12
Topic 9.1 — Line–Line
Question 1
Lines \(L_1: (1,2)+t(3,1)\) and \(L_2:(0,5)+s(1,-1)\) in R². They intersect at a point. The y-coordinate of the intersection is:
Solution:
\(1+3t=s\) and \(2+t=5-s\). From first: \(s=1+3t\). Sub: \(2+t=5-(1+3t)=4-3t\) → \(4t=2\) → \(t=0.5\). Point: \((2.5, 2.5)\). Hmm: \(y=2+0.5=2.5\). Let me recheck: From line 2 with \(s=1+3(0.5)=2.5\): point \((0+2.5,5-2.5)=(2.5,2.5)\). So y = 2.5.
Question 2
Two lines in R³ are skew if:
Solution:
Skew = non-parallel AND non-intersecting (must be in 3-D).
Topic 9.2 — Line–Plane
Question 3
Find \(t\) where line \((x,y,z)=(1,2,3)+t(1,1,1)\) meets plane \(x+y+z=12\).
Solution:
Sub: \((1+t)+(2+t)+(3+t)=12\) → \(6+3t=12\) → \(t=2\).
Question 4
Same line and \(t=2\): the intersection point is \((3, ?, 5)\). Find ? .
Solution:
\(y=2+2=4\).
Question 5
Line \((x,y,z)=(1,1,1)+t(2,-1,3)\) and plane \(2x-y+3z=4\). The line is:
Solution:
\(\vec d\cdot\vec n=4+1+9=14\ne 0\) → not parallel to plane → exactly one intersection.
Topic 9.3 — Plane–Plane
Question 6
Planes \(\pi_1: x+y+z=3\) and \(\pi_2: x-y+z=1\). The line of intersection has direction \(\vec n_1\times\vec n_2 = (a,0,b)\). Find \(a\).
Solution:
\((1,1,1)\times(1,-1,1)=(1\cdot1-1\cdot(-1),\,1\cdot1-1\cdot1,\,1\cdot(-1)-1\cdot1)=(2,0,-2)\). \(a=2\).
Question 7
A point on the line of intersection of \(x+y+z=3\) and \(x-y+z=1\) (set \(z=0\)). Find \(x\).
Solution:
System with \(z=0\): \(x+y=3,\,x-y=1\) → \(x=2,y=1\).
Topic 9.4 — Three Planes
Question 8
Three distinct planes can intersect in:
Solution:
Configurations include unique point, line (consistent dependent), plane (all same), or empty (parallel/triangular prism).
Question 9
Solve: \(x+y+z=6,\ 2x-y+z=3,\ x+2y-z=2\). Find \(x\).
Solution:
Add (1)+(3): \(2x+3y=8\). (1)-(2): \(-x+2y=3\) → \(x=2y-3\). Sub: \(2(2y-3)+3y=8\) → \(7y=14\) → \(y=2\), \(x=1\). From (1): \(z=3\).
Question 10
For the system in Q9, the value of \(z\) is:
Solution:
\(z=6-1-2=3\).
Question 11
Two lines have direction vectors that are not parallel. In R³, they could be:
Solution:
Non-parallel lines in R³ may intersect or be skew (no common point).
Question 12
Three planes whose normals are coplanar (linearly dependent) but constants don't match — the system is:
Solution:
Triangular-prism type: no common intersection.

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