Diagnostic — checks Units 6–7 + linear-equation prereqs
📋 Not Graded
Purpose: Probes vector operations, dot/cross products, and linear-equation algebra needed for line/plane equations.
Q1 Prereq — Vectors
[3]
Given \(\vec u=(2,-3)\) and \(\vec v=(3,2)\): a) compute \(\vec u\cdot\vec v\); b) state whether they are perpendicular; c) state a vector parallel to \(\vec u\) of length 1.
Q2 Prereq — Linear algebra
[3]
Solve the system: \(2x+y=7,\ x-y=2\). Then write the solution as a position vector \((x,y)\).
Q3 Prereq — Cross product
[2]
Compute \((1,0,2)\times(3,1,0)\). What special property does this result have relative to the two input vectors?
Q4 Thinking
[3]
A plane in 3-D can be specified using either: (i) one point and a normal vector, or (ii) one point and two non-parallel direction vectors. Sketch (in words) why both methods uniquely determine the plane.
Q5 Communication
[3]
Explain in your own words the difference between vector, parametric, and symmetric forms of a line equation. Which can represent a line in R²? In R³?
Q6 Thinking
[3]
Why is there NO single scalar (Cartesian) equation for a line in R³, but there IS one for a plane in R³? Explain.
Q7 Reflection
[2]
3-D geometry is challenging. What helps you visualize 3-D? (Models, software, drawing axes, …)?
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