Vectors

• A vector equation of a straight line passing through the point A with position vector a, and parallel to vector b is:

  
  

  r = a +λb
• A vector equation of a straight line passing through points C and D, with position vectors c and d is:

  
  

  r = c + λ(d - c)
• If a = (a1, a2, a3) and b = (b1, b2, b3), the equation of the line with the vector equation r = a + λb can be given in cartesian form by:

  
  

  (Each of these equations is equal to λ)

  
• The vector equation of a plane is r = a + λb + μc, where:

  • r is the position vector of a general point in the plane
  • a is the position vector of a point in the plane
  • b and c are non-parallel, non-zero vectors in the plane
  • λ and μ are scalars
• A cartesian equation of a plane in three dimensions can be written in the form az + by + cz = z, where a, b, c and d are constants and vector (a, b, c) is the normal vector to the plane
• The scalar product of two vectors a and b is written a.b, and defined as:

  (where θ is the angle between a and b)

  
• If a and b are the position vectors of the points A and B then:

  
• The non-zero vectors a and b are perpendicular only if a.b = 0
• If a and b are parallel then:

  
• To find the scalar product of two three by one vectors:

  
• The acute angle between two intersecting straight lines, where a and b are direction vectors of the lines, is given by:

  
• The scalar product form of an equation of a plane is r.n = k where k = a.n for any point in the plane with position vector a
• The acute angle between the line with the equation r = a + λb and the plane with equation r.n = k is given by the formula:

  
• The acute angle between the plane with equation r.n1 = k1 and the plane with equation r.n2 = k2 is given by the formula:

  
• Two lines are skew if they aren't parallel and they do not intersect
• For any two non-intersecting lines, there is a unique line segment AB such that A lies on the first and B lies on the second and AB is perpendicular to both
• The perpendicular point from point P to line l is drawn from P at right angles to l
• The perpendicular from a point P to a plane is a line drawn from P parallel to the normal vector n