MCV4U – Grade 12 Calculus & Vectors – Equations of Lines and Planes

Grade 12 – Calculus and Vectors

Lines and Planes Test

 Equation of 2-Space and 3-Space lines

  • In 2-Space, a line can be defined in 4 ways: Slope y-intercept form, vector equation, parametric form, or scalar form
    • Slope Y-Intercept form: y = mx + b
      • m is the slope
      • b is the y intercept
    • Vector Form: r = r0 + tm
      • or: [x , y] = [x0 , y0] + t[m1 , m2]
      • [x,y] is a position vector to any point on the line
      • [x0 , y0] is a position vector to any point on the line
      • [m1 , m 2] is the direction vector for the line
    • Parametric Form: 
      • x = xo + tm1 or x = x0 + t[m1]
      • y = y0 + tm2 or y = y0 + t[m2]
      • Same definitions as Vector form, derived from vector form
    • Scalar Form: ax + by + c = 0
      • n = [a, b] is a normal vector to the line
      • If direction vector d = [a, b], then normal vector n = [-b, a]

 

  • In 3-Space, a line can be defined by a vector equation or parametric equation
    • Vector Form: r = r0 + tm
      • or: [x , y , z] = [x0 , y0 , z0] + t[m1 , m2 , m3]
      • [x,y,z] is a position vector to any point on the line
      • [x0 , y0 , z0] is a position vector to any point on the line
      • [m1 , m2 , m3] is the direction vector for the line
    • Parametric Form: 
      • x = xo + tm1 or x = x0 + t[m1]
      • y = y0 + tm2 or y = y0 + t[m2]
      • z = z0 + tm3 or y = z0 + t[m3]
      • Same definitions as Vector form, derived from vector form
  • A normal vector is a line perpendicular to that line
  • To determine a line, 2 points, or a point and a direction vector are needed
  • For lines in 2-Space, a point and the normal vector can also determine the line

 

Equations of Planes

  • In 2-Space, a scalar equation defines a line. In 3-space, a scalar equation defines a plane
  • In 3-Space, a plane can be defined by a vector equation, parametric equation, or a scalar equation
    • Vector Form: r = r0 + tm
      • or: [x , y , z] = [x0 , y0 , z0] + t[a1 , a2 , a3] + s[b1 , b2 , b3]
      • [x,y,z] is a position vector to any point on the plane
      • [x0 , y0 , z0] is a position vector to any point on the plane
      • [a1 , a2 , a3] and [b1 , b2 , b3] have to be non-parallel vectors on the plane
    • Parametric Form: 
      • x = xo + ta1 + tb1    or x = x0 + t[a1] + s[b1]
      • y = y0 + ta2 + tb2   or y = y0 + t[a2] + s[b2]
      • z = z0 + ta3 + tb3   or y = z0 + t[a3] + s[b3]
      • Same definitions as Vector form, derived from vector form
    • Scalar Form: ax + by + cz + d = 0
      • Where [a,b,c] is a vector normal to the plane
  • A plane can be defined by 3 non-collinear points or a point and 2 non-parallel direction vectors
  • Intercepts of a plane
    • The X-Intercept of a plane is found by setting y = z = 0 and solving for x
    • The Y and Z intercepts are the same by setting the other 2 variables to zero

 

Intersection of Lines in 2-Space and 3-Space

  • In 2-Space, the intersection of 2 lines have 3 possibilities.
    • (1) Lines intersect at exactly one point
    • (2) Lines are coincident, so there are infinitely many solutions
    • (3) Lines are parallel, no solutions
  • In 3-Space, the intersection of 2 lines can have 4 possibilities
    • (1) They intersect at one point
    • (2) They are coincident
    • (3) They are parallel
    • (4) They are skewed lines, where they are not parallel and have no solution
  • To find the distance between 2 skew lines, the formula:
    • d = | (P1P2 dot n / n) |
    • where P1 and P2 are any points on each line and n = m1 x m2.
  • To find the intersection, simply equate each pair of lines in parametric form and solve for X, Y, and Z in each set.

 

Intersection of Planes and Lines

  • In 3-Space, the intersection of planes and lines have 3 possibilities
    • (1) The line intersects right through the plane at one point
    • (2) The line is coincident with the plane at infinitely many points
    • (3) The line is parallel and not touching the plane with no solutions
  • The distance between between point P and a plane is:
    • d = |n dot PQ| / |n|
      • where n is a normal vector to the plane, Q is any point on the plane
  • To find the intersection, substitute each X, Y and Z of the parametric equation into the equation of the plane. Then after t is found, substitute that back into the line to find what each value of X Y and Z are.

 

Intersection of Planes

  • There are 3 possibilities for the intersection of 2 planes
    • (1) The planes intersect at a line
    • (2) The planes are coincident, with infinitely many solutions
    • (3) The planes are parallel with no solutions
  • With 2 planes:
    • 1) Check if the normals are parallel
    • 2) Try solving based on whether or not the normals are parallel
      • If they are parallel, you’ll get no or infinite solutions
      • If they aren’t parallel, you’ll get a line as the solution

 

  • There are 7 possibilities for intersection of 3 planes:
    • If their normals are parallel
      • (1) All 3 planes are coincident if normals are all the same
      • (2) All 3 planes are parallel if normals are all distinct
      • (3) 2 Planes are coincident and 1 is parallel if 2 normals are distinct
      • (4) 1 Plane intersects 2 planes that are coincident or parallel if 2 normals are the same
    • If their normals are not parallel
      • If triple scalar product (n1 x n2)*n3 = 0
        • (5) A Triangular prism is formed and no intersection (0x + 0y = c)
        • (6) A “revolving door” is formed with a line intersection (0x + 0y = 0)
      • If triple scalar product is not = 0
        • (7) All planes intersect at a point
  • For revolving door intersection and intersection of 2 planes at a line, a parameter z = t should be used to find the line of intersection.