Grade 12 Calculus Chapter 8

Normal vector

- A normal vector to a line l is a vector n that is perpendicular to the line. - If l: Ax + By + C = 0, then vector n = [A,B].

Procedures to determine if the line and the plane intersect (Method 2, using vectors):

- Determine the direction vector m of the line - Determine the normal vector n of the plane. - If the dot product doesn't equal to 0, meaning m and n are not perpendicular. Thus the plane and the line are not parallel or coincident, they must intersect. - If the dot product equals to 0, meaning the plane and the line are parallel to each other. - Substitute a point on the line into the plane, if LS=RS, then they are coincident. If not, they are distinct.

Procedures to determine the shortest distance between a line and a plane (if they are not parallel):

- Determine the normal vector [A, B, C] of the plane Ax + By + Cz + D = 0. - Choose any point, Q, on the plane. - Determine vector PQ, which P is a point on the line. - The projection of vector PQ on the normal vector [A, B, C] is the distance.

Procedures to determine the distance between these lines is approximately 1.15 units.

- First get two points, A and B, from vector equations of two lines - Determine vector AB - Use the direction vectors for both lines to calculate the normal vector n to the lines. - The shortest distance between two skew lines is their length of common perpendicular, which equals the projection of AB on normal vector n.

Procedures to determine if the line and the plane intersect (Method 1):

- Substitute the parametric equations into the scalar equation of the plane, which gives you a t value. - Substitute this t value back into parametric equations, gives the intersection point. - If the result is 0t = 0, then there are infinite solutions. - If the result is 0t = a real number, then no solution.

3 possibilities for the intersection of a line and a plane:

- The line and the plane intersect at a point. There is exactly one solution. - The line lies on the plane. An infinite number of solutions. - The lie is parallel to the plane. No solutions.

3 possibilities for the intersection of two lines:

- The lines intersect at a point, so there is exactly one solution. - The lines are coincident, so there are infinitely many solutions. - The lines are parallel, so there is no solution.

4 possibilities for the intersection of two lines:

- The lines intersect at a point, so there is exactly one solution. - The lines are coincident, so there are infinitely many solutions. - The lines are parallel, sol there is no solution. - The lines are skew (neither parallel nor intersect).

3 possibilities that 3 planes have no intersections:

- The planes are parallel and at least two are distinct. - Two planes are parallel and the third intersects both of the parallel planes. - The planes intersect in pairs.

3 possibilities for the intersection of 3 planes:

- The planes intersect in a line. An infinite number of solutions. - The planes are coincident. There are an infinite number for solutions. - The planes are parallel and distinct. There is no solution.

Parametric equations

- The vector equation of a line can be separated into two parts, one for each variable. These are called the parametric equations of the line, because the result is governed by the parameter t, t belongs to R. -The vector equation r = r0 + tm can be written [x, y] = [x0, y0] + t[m1, m2]

Procedures to check if two lines in three-space intersect (If they are parallel) (Method 2)

- Write each equation in parametric form. - Rearrange equation 1 to isolate s. - Substitute s = t -4 into equation 2 and 3 and solve for t. - 0t = 0 has no solutions. The two lines are parallel and distinct.

Procedures to check if two lines in three-space intersect (If they are not parallel)

- Write the equations in parametric form. - Equate expressions for like coordinates. - Use equations 2 and 3 for s and t. - Check that this pair of s and t satisfy the first equation. - If left side equals right side, the nthis system has a unique solution at (1, 1, 3).

Procedures to check if two lines in three-space intersect (If they are parallel) (Method 1)

- Write the parametric equations for l2. - Substitute the a pair of coordinates on l2 (usually [x0, y0, z0]). - Solve for 5 values in those three equations. - If t are equal then two lines intersect, if not then they have no intersections.

Procedures to check if two lines in three-space are skew lines.

- Write them in parametric form - Equate the expressions for each variable. - Solve 2 and 3 for s and t. - Check if left side equals right side in equation 1. - If not, then they are skew lines.

Vector equation vol.2

- r = r0 + tm, t belongs to real numbers. - vector r = [x,y] is a position vector to any unknown point on the line. - vector r0 = [x0, y0] is a position vector to any known point on the line. - vector m = [m1, m2] is a direction vector parallel to the line.

There are three ways that 3 planes can intersect:

-The planes intersect at a point. There is exactly one solution. (The normals are not parallel and not coplanar) - The planes intersect in a line. There are an infinite number of solutions. (The normals are not parallel and but coplanar) - The planes are coincident, their normals are parallel.

Direction vector

A vector parallel to a line or plane.

Vector equation of a plane

An equation of the form [x, y, z] = (x0, y0, z0) + t[m1, m2, m3] + s[b1, b2, b3], where P0(x0, y0, z0) is a point on the line and vector a = [a1, a2, a3] and vector b = [b1, b2, b3] are non-collinear direction vectors for the plane.

Scalar equation of a line

An equation of the form [x, y, z] = (x0, y0, z0) + t[m1, m2, m3], where P0(x0, y0, z0) is a point on the line and vector m = [m1, m2, m3] is a direction vector for the line.

Skew lines

Straight lines in 3-D that are neither parallel nor intersecting.