Line and Plane — Class 12 Maharashtra HSC Mathematics revision notes
A 5-page visual revision sheet for Line and Plane, written for Class 12 Maharashtra HSC Mathematics. Formulas, diagrams and the traps that cost marks, laid out to be read the night before rather than the term before. Free, and it opens inside CapCompass.
Page 1 as PDF
Line and Plane in one look
a point and a direction build a line; a point and a normal build a plane
Vector Equation of a Line
- a line is fixed by a point and a direction: through A(a) parallel to b, AP=lambdab
- the PARAMETRIC form: r=a+lambdab. λ is the parameter, each real value gives one point
- the same line, no parameter: (r-a)timesb=0, the non-parametric form
- through TWO points A(a), B(b): r=a+λ(b-a), since b-a is a direction vector
Cartesian Forms of a Line
- through (x_1,y_1,z_1) with direction ratios a,b,c
x-x_1/a=y-y_1/b=z-z_1/c - through two points (x_1,y_1,z_1) and (x_2,y_2,z_2)
x-x_1/x_2-x_1=y-y_1/y_2-y_1=z-z_1/z_2-z_1 - parametric form, from the symmetric form set equal to λ
x=x_1+λ a, y=y_1+λ b, z=z_1+λ c
Vector and Cartesian equations of a line
Q. find the vector and Cartesian equations of the line through (1,2,3) with direction ratios 2,-1,2
- vector form: r=a+lambdab with a=i+2j+3k and b=2i-j+2k
- r=(i+2j+3k)+λ(2i-j+2k)
- Cartesian: x-1/2=y-2/-1=z-3/2
Answer: r=(i+2j+3k)+λ(2i-j+2k),quad x-1/2=y-2/-1=z-3/2
Distance of a Point from a Line
- line r=a+lambdab, point P(α) off it: drop perpendicular PM, M is AP's projection onto b
- PM is found from right triangle AMP: PM^2=AP^2-AM^2, AM the scalar projection of α-a along b
Distance-from-a-Line Formula
- distance of P(α) from the line r=a+lambdab
PM=√|α-a|^2-left(frac(α-a)cdotb|b|right)^2
Page 1 of 5. The full sheet, all 5 pages, opens free in the app.
What this Line and Plane sheet covers
- Vector Equation of a Line
- Cartesian Forms of a Line
- through (x_1,y_1,z_1) with direction ratios a,b,c
- through two points (x_1,y_1,z_1) and (x_2,y_2,z_2)
- parametric form, from the symmetric form set equal to λ
- Vector and Cartesian equations of a line
- Distance of a Point from a Line
- Distance-from-a-Line Formula
- distance of P(α) from the line r=a+lambdab
- Skew Lines
- Two lines, two very different pictures
- Shortest Distance Formulas
- between lines r=a_1+lambdab_1 and r=a_2+mub_2 (skew case)
- the two lines INTERSECT exactly when (assumes they are not parallel)
- between two PARALLEL lines r=a_1+lambdab and r=a_2+lambdab
- Shortest distance between skew lines
- Things to remember before the plane starts
- Line vs plane: what fixes each
- Line
- Plane
- Equation of a Plane
- Plane Equations, One Family
- point and normal, vector form
- point and normal, Cartesian form, (x_1,y_1,z_1) with normal ratios a,b,c
- point and two parallel vectors
- through three non-collinear points
- Degenerate and special cases
- Normal Form of a Plane
- at distance p from the origin, unit normal n
- Cartesian normal form, l,m,n the direction cosines of the normal
- foot of the perpendicular from the origin, ON=p
- Plane through a point, normal to a vector
- Angle Between Planes, and Between a Line and a Plane
- Angle Formulas
- angle between planes rcdotn_1=d_1 and rcdotn_2=d_2
- angle between line r=a+lambdab and plane rcdotn=d
- Angle between two planes
- Coplanarity of Two Lines
- lines r=a_1+lambdab_1 and r=a_2+mub_2 are coplanar iff
- the plane they determine, once coplanar
- Distance of a Point from a Plane
- Point-to-Plane Distance
- distance of A(a) from the plane rcdotn=p, unit normal
- Symbols
- Traps that cost marks
- Tricks that save time
How CapCompass uses this sheet
These notes are not a standalone PDF. CapCompass plans Mathematics chapter by chapter against your real exam dates and free hours, and when the plan reaches Line and Plane the sheet opens inside the study block itself — at the page covering the sub-topic that block is on, with the timer still running. Revision arrives at the moment the plan says revise, instead of having to be gone looking for.