PYQ Vault
EbooksTutor

MH HSC 12 Maths

Line and Plane

Each concept is a short lesson, then questions until you get three right.

0/11

11 concepts, a short lesson each

  • Lines0/5
  • Planes0/6

Start here

Vector and Cartesian equations of a line

Through a⃗\vec a along b⃗\vec b: r⃗=a⃗+λb⃗\vec r = \vec a + \lambda\vec b, or x−x1a=y−y1b=z−z1c\dfrac{x - x_1}{a} = \dfrac{y - y_1}{b} = \dfrac{z - z_1}{c}; through two points, take b⃗=B⃗−A⃗\vec b = \vec B - \vec A

Start

Lines0 of 5

  1. Vector and Cartesian equations of a lineThrough a⃗\vec a along b⃗\vec b: r⃗=a⃗+λb⃗\vec r = \vec a + \lambda\vec b, or x−x1a=y−y1b=z−z1c\dfrac{x - x_1}{a} = \dfrac{y - y_1}{b} = \dfrac{z - z_1}{c}; through two points, take b⃗=B⃗−A⃗\vec b = \vec B - \vec AStart
  2. Angle between two linescos⁡θ=∣b⃗1⋅b⃗2∣∣b⃗1∣∣b⃗2∣\cos\theta = \dfrac{|\vec b_1\cdot\vec b_2|}{|\vec b_1||\vec b_2|} from the direction vectors; perpendicular when a1a2+b1b2+c1c2=0a_1a_2 + b_1b_2 + c_1c_2 = 0Start
  3. Distance of a point from a lineWrite the foot as a⃗+λb⃗\vec a + \lambda\vec b, make PM→⋅b⃗=0\overrightarrow{PM}\cdot\vec b = 0 to find λ\lambda, then measure PMPMStart
  4. Shortest distance between two linesSkew lines: d=∣(a⃗2−a⃗1)⋅(b⃗1×b⃗2)∣∣b⃗1×b⃗2∣d = \dfrac{|(\vec a_2 - \vec a_1)\cdot(\vec b_1\times\vec b_2)|}{|\vec b_1\times\vec b_2|}; parallel lines: d=∣(a⃗2−a⃗1)×b⃗∣∣b⃗∣d = \dfrac{|(\vec a_2 - \vec a_1)\times\vec b|}{|\vec b|}Start
  5. Intersecting and coplanar linesTwo lines are coplanar (meet or are parallel) when (a⃗2−a⃗1)⋅(b⃗1×b⃗2)=0(\vec a_2 - \vec a_1)\cdot(\vec b_1\times\vec b_2) = 0; find the meeting point by equating the parametersStart

Planes0 of 6

  1. Equation of a plane from a normalThrough a⃗\vec a with normal n⃗\vec n: r⃗⋅n⃗=a⃗⋅n⃗\vec r\cdot\vec n = \vec a\cdot\vec n, i.e. ax+by+cz=dax + by + cz = d; normal form r⃗⋅n^=p\vec r\cdot\hat n = p with pp the distance from OStart
  2. Plane through three points or containing linesTake the normal as a cross product of two directions in the plane, or set the determinant of r⃗−a⃗\vec r - \vec a and the two directions to 00Start
  3. Foot of the perpendicular and interceptsThe foot from O to ax+by+cz=dax + by + cz = d is da2+b2+c2(a,b,c)\dfrac{d}{a^2 + b^2 + c^2}(a, b, c); intercept form xa+yb+zc=1\dfrac{x}{a} + \dfrac{y}{b} + \dfrac{z}{c} = 1Start
  4. Distance of a point from a planed=∣ax1+by1+cz1−d∣a2+b2+c2d = \dfrac{|ax_1 + by_1 + cz_1 - d|}{\sqrt{a^2 + b^2 + c^2}}, or ∣p⃗⋅n⃗−d∣∣n⃗∣\dfrac{|\vec p\cdot\vec n - d|}{|\vec n|} in vector formStart
  5. Angle between planes, and a line and a planeBetween planes, the angle between their normals; between a line and a plane, sin⁡θ=∣b⃗⋅n⃗∣∣b⃗∣∣n⃗∣\sin\theta = \dfrac{|\vec b\cdot\vec n|}{|\vec b||\vec n|}Start
  6. A line meeting or lying in a planePut the line's general point into the plane to find where they meet; the line lies in the plane when b⃗⋅n⃗=0\vec b\cdot\vec n = 0 and one point satisfies itStart

Practise Line and Plane

Once the concepts click, try the questions your exam asks.

Textbook exercisesEvery exercise, with model answersPast board questionsAsked from this chapter, with solutions