MCV4U - Equations of Lines and Planes

MCV4U – Grade 12 Calculus & Vectors – Equations of Lines and Planes Equation of 2-Space and 3-Space lines ● There are four ways to define a line in two dimensions: scalar form, vector equation, slope y-intercept form, and parametric 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 of 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] = [x 0 , y0 , z0] + t[a1 , a2 , a3] + s[b1 , b 2 , b 3] ■ [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 nonparallel 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 has 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.