📝 Unit 8: Lines and Planes

Vector · Parametric · Symmetric · Scalar Equations
🔄 Not Graded
Purpose: Self-check on equations of lines and planes in R² and R³.
Score: 0 / 12
Topic 8.1 — Lines in R²
Question 1
Vector equation of line through \((2,1)\) with direction \((3,-1)\). At \(t=2\), point is:
Solution:
\((x,y)=(2,1)+t(3,-1)\); at \(t=2\): \((8,-1)\). x = 8.
Question 2
Scalar (Cartesian) equation of line through \((1,2)\) perpendicular to \(\vec n=(3,4)\):
Solution:
\(\vec n\cdot(\vec r-\vec r_0)=0\): \(3(x-1)+4(y-2)=0\) → \(3x+4y=11\).
Topic 8.2 — Lines in R³
Question 3
Symmetric equations of line through \((1,2,3)\) with direction \((4,5,6)\). The denominator under \(y-2\) is:
Solution:
\(\frac{x-1}{4}=\frac{y-2}{5}=\frac{z-3}{6}\). Denominator = 5.
Question 4
Parametric equation of line through \((0,1,-2)\) parallel to \((3,-2,1)\). Find \(z\) when \(t=2\).
Solution:
\(z=-2+t\); at \(t=2\): \(0\).
Question 5
Two lines have direction vectors \((1,2,1)\) and \((2,4,2)\). They are:
Solution:
Second is 2× first → parallel directions.
Topic 8.3 — Planes
Question 6
Scalar equation of plane through \((1,2,3)\) with normal \(\vec n=(2,-1,4)\). The constant term \(D\) in \(2x-y+4z=D\) is:
Solution:
\(2(1)-2+4(3)=2-2+12=12\).
Question 7
Normal vector to plane \(3x-2y+z-7=0\) is:
Solution:
Coefficients of \(x,y,z\): \((3,-2,1)\).
Question 8
Plane through \((1,0,0),(0,2,0),(0,0,3)\). The constant in \(\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1\) form gives \(a+b+c=\)
Solution:
Intercepts: \(a=1,b=2,c=3\). Sum = 6.
Topic 8.4 — Distance Formulas
Question 9
Distance from \((1,2,3)\) to plane \(x+2y+2z=12\). (Use \(\frac{|ax_0+by_0+cz_0-d|}{\sqrt{a^2+b^2+c^2}}\).)
Solution:
\(|1+4+6-12|/\sqrt{1+4+4}=|{-1}|/3=1/3\approx 0.333\).
Question 10
Distance from origin to line in R² with equation \(3x+4y=10\):
Solution:
\(|0+0-10|/\sqrt{9+16}=10/5=2\).
Question 11
A plane has equation \(x-y+2z=6\). The point on it with \(y=z=0\) has \(x=\)
Solution:
\(x=6\).
Question 12
Two planes \(2x+y-z=5\) and \(4x+2y-2z=7\). They are:
Solution:
Normals \((2,1,-1)\) and \((4,2,-2)\) are parallel; constants \(5\ne 7/2\) → distinct parallel.

📊 Self-Reflection

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