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 = r**_{0}**+ tm**- or:
**[x , y] = [x**_{0}**, y**_{0}**] + t[m**_{1}**, m**_{2}**]** - [x,y] is a position vector to any point on the line
- [x
_{0}, y_{0}] is a position vector to any point on the line - [m
_{1}, m_{2}] is the direction vector for the line **Parametric Form:****x = x**_{o}**+ tm**_{1 }**or x = x**_{0}**+ t[m**_{1}**]****y = y**_{0}**+ tm**_{2}**or y = y**_{0}**+ t[m**_{2}**]**- 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 = r**_{0}**+ tm**- or:
**[x , y , z] = [x**_{0}**, y**_{0 , }**z**_{0}**] + t[m**_{1}**, m**_{2 , }**m**_{3}**]** - [x,y,z] is a position vector to any point on the line
- [x
_{0}, y_{0 , }z_{0}] is a position vector to any point on the line - [m
_{1}, m_{2 , }m_{3}] is the direction vector for the line **Parametric Form:****x = x**_{o}**+ tm**_{1 }**or x = x**_{0}**+ t[m**_{1}**]****y = y**_{0}**+ tm**_{2}**or y = y**_{0}**+ t[m**_{2}**]****z = z**_{0}**+ tm**_{3}**or y = z**_{0}**+ t[m**_{3}**]**- 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 = r**_{0}**+ tm**- or:
**[x , y , z] = [x**_{0}**, y**_{0 , }**z**_{0}**] + t[a**_{1}**, a**_{2 , }**a**_{3}**] + s[b**_{1}**, b**_{2 , }**b**_{3}**]** - [x,y,z] is a position vector to any point on the plane
- [x
_{0}, y_{0 , }z_{0}] is a position vector to any point on the plane - [a
_{1}, a_{2 , }a_{3}] and [b_{1}, b_{2 , }b_{3}] have to be non-parallel vectors on the plane **Parametric Form:****x = x**_{o}**+ ta**_{1}**+ tb**_{1 }**or x = x**_{0}**+ t[a**_{1}**] + s[b**_{1}**]****y = y**_{0}**+ ta**_{2}**+ tb**_{2}**or y = y**_{0}**+ t[a**_{2}**] + s[b**_{2}**]****z = z**_{0}**+ ta**_{3}**+ tb**_{3 }**or y = z**_{0}**+ t[a**_{3}**] + s[b**_{3}**]**- 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 = | (P**_{1}**P**_{2}**dot n / n) |**- where P
_{1}and P_{2 }are any points on each line and n = m_{1}x m_{2}. - 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.