Diagnostic — checks Unit 8 line/plane equations + linear systems prereqs
📋 Not Graded
Purpose: Probes confidence with line/plane equations from Unit 8 and 3×3 linear systems algebra needed for Unit 9.
Q1 Prereq — Unit 8
[3]
Write parametric equations for the line through \((2,1,3)\) parallel to \((1,-2,4)\).
Q2 Prereq — Linear systems
[3]
Solve the 2×2 system \(2x+y=7,\ x-2y=-4\). Then solve the 3×3 system \(x+y+z=4,\ x-y=0,\ z=2\).
Q3 Prereq — Substitution
[2]
If a parametric line is \(x=1+2t,\,y=3-t,\,z=t\), substitute into the equation \(x+2y+z=10\) and solve for \(t\).
Q4 Thinking
[3]
Two lines in R³ may be: parallel (same direction, no common point), parallel-coincident (same line), intersecting in one point, or skew. Briefly describe each case and the test you'd perform first to distinguish them.
Q5 Communication
[3]
Explain in your own words why solving the intersection of two planes is equivalent to solving 2 equations in 3 unknowns. Why does this generally yield a one-parameter family (a line)?
Q6 Thinking
[3]
Three planes can: (a) meet at a single point, (b) meet along a line, (c) be coincident, or (d) have no common point. Sketch (in words) one example of each.
Q7 Reflection
[2]
Solving 3×3 linear systems can be tedious. Which method do you prefer (substitution, elimination, matrix)? Why?
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