1. Given vectors a, b, and c: Graph: a – b + 2c and 3c – 2a + b 2. Prove that these following vectors a = 3i – 2j + k, b = i – 3j +5k, and c = 2i +j – 4k formed a right triangle

Line Equation How to determine the equation of l passing through P ( x 0,y 0,z 0 ) parallel a vector v? Let Q ( x,y,z ) is an arbitrary point on l, suppose r 0 and r are positional vetors posisi of P and Q. If a is the representation of The rule of the addition of vectors defined: x y z v a r0r0 r P(x0,y0,z0)P(x0,y0,z0) Q(x,y,z) r = r 0 + a Because a and v are parallel, there exist t so that a = t v r = r 0 + t v l The equation of line vector

If v =  a, b, c , r =  x, y, z  and r 0 =  x 0, y 0, z 0 , then the implication of the equation is x= x 0 + ta, y = y 0 + tb, z = z 0 + tc which is called parametric equation of a line passing through P(x 0, y 0, z 0 ) by vector v =  a, b, c . By solving t from the parametric equation, we get Which is called symmetrical equation of a line passing through P(x 0, y 0, z 0 ) by vector v =  a, b, c .

Solve in group: 1.Determine the equation of a line passing through P(5, 1, 3) which has the same direction with vector v = 3i – 5j + 2k. Determine other two points on that line. 2.Determine that these following lines are skew lines (do not intersect each other): x = 1 + ty = -2 +3tz = 4 – t x = 2sy = 3 + sz = s 1.Determine the equation of a plane passing through P(2,4,-1) with n =  2,3,4  as the normal vector. Determine the intersecting point with the axis. 2.Determine the equation of a plane passing through P(1,3,2), Q(3,-1,6), and R(5,2,0). 3.Determine the intersecting point of lines x = 2 + 3t, y = -4t, z = 5 + t which intersect 4x + 5y – 2z = 18.

E.g: 1.Determine the equation of a line passing through P(5, 1, 3) which has the same direction with vector v = 3i – 5j + 2k. Determine other two points on that line. 2.Determine the equation of a line passing through P(2, 4, -3) and Q(3, -1, 1). Determine the intersecting point of the line and plane -xy? Determine the intersecting point of the line and plane x – 2y + 3z = 5. 3.Determine that these following lines are skew lines (do not intersect each other): x = 1 + ty = -2 +3tz = 4 – t x = 2sy = 3 + sz = s

Plane equation A plane in three dimensions determined by a point P(x 0, y 0, z 0 ) and a vector n which is perpendicular to the plane (normal vector). Let Q ( x,y,z ) is an arbitrary point in a plane, let r 0 and r are positional vectors of P and Q. Vector r – r 0 can be determined by Normal vector n perpendicular to every vector on that plane, particularly r – r 0 therefore x y z n n  (r – r 0 ) = 0 P(x0,y0,z0)P(x0,y0,z0) Q(x,y,z) r0r0 r r – r 0 The equation of vector from a plane

If n =  a, b, c , r =  x, y, z  and r 0 =  x 0, y 0, z 0 , therefore the equation can be: a(x – x 0 ) + b(y – y 0 ) + c(z – z 0 ) = 0 This equation is called scalar equation of a plane passing through P(x 0, y 0, z 0 ) with n =  a, b, c  as the normal vector. The equation can be written as this following linear equation. ax + by + cz + d = 0

E.g: 1.Determine the equation of a plane passing through P(2,4,-1) with n =  2,3,4  as the normal vector. Determine the intersecting point with the axis. 2.Determine the equation of a plane passing through P(1,3,2), Q(3,-1,6), and R(5,2,0). 3.Determine the intersecting point of lines x = 2 + 3t, y = -4t, z = 5 + t which intersect 4x + 5y – 2z = Determine the angle between x + y + z = 1 and x – 2y + 3z = 1. And, determine the equation of intersecting lines of those two planes.

5.Determine the formula to determine the distance from Q(x 1,y 1,z 1 ) to ax + by + cz + d = 0. 6.Determine the distance between these following two parallel planes 10x + 2y – 2z = 5 and 5x + y – z =1. 7.Determine the distance between these following two lines x = 1 + ty = -2 +3tz = 4 – t x = 2sy = 3 + sz = s P(x0,y0,z0)P(x0,y0,z0) Q(x 1,y 1,z 1 ) b n