Vectors

• The equation of a line in vector form is r = p + t d where p is a position vector, d is a direction vector and t is a scalar parameter.
• The equation of a line in cartesian form is (x-pi)/di = (y-pj)/dj = (z-pk)/dk where p is position vector (pi pj pk) and d is direction vector (di dj dk).
• The angle between two vectors is the angle between them when both are pointed away from their point of intersection.
• The scalar dot product of vectors a and b is denoted a.b and is (ai x bi)+(aj x bj)+(ak x bk).
• The angle between two vectors a and b is a.b = |a| |b| cos theta.
• Two vectors are perpendicular if their scalar product equals zero.
• Two vectors are perpendicular if their scalar product equals zero.
• The equation of a plane in vector form is r = p + s d1 + t d2 where p is a position vector, d is a direction vector and s and t are scalar parameters.
• The equation of a plane in cartesian form is ni x + nj y + nk z = d where n is a normal vector and d is the scalar product of the normal vector and a point on the plane. Can also be expressed in a simplified form as r.n = d.
• The angle between the vector p + t d and the plane with normal vector n is d.n = |d| |n| sin theta.
• The angle between two planes with normal vectors n1 and n2 is n1.n2 = |n1| |n2| cos theta.
• Vectors are parallel if their direction vectors are multiples of each other.
• If vectors intersect there will be a common value of the scalar parameters that satisfies all three equations when the i, j and k components of both vectors are equated.
• Vectors are skew if they do not intersect and are not parallel.
• To find the intersection of a line and a plane, express the plane in simplified cartesian form and replace general point r with the equation of the line to get l.n = d. Then solve the resultant expression.
• The shortest distance between vectors is when the line between them is perpendicular to both.
• To find the equation of the line between a point and a line, subtract the point from the general equation of a line.
• To find the equation of the line between two lines, subtract one general equation of the line from the other.
• The shortest line between a point p and a line l can be found by finding the scalar product of the vector between p and l and the direction vector of l and equal it to zero in order to find the specific scalar parameter of l.
• The shortest line between two lines l1 and l2 can be found by finding the scalar product of the vector between l1 and l2 and the direction vectors of both lines and equal it to zero in order to form simultaneous equations of the specific scalar parameters of l1 and l2.
• The equation of the line between a point and its reflected point in a plane is r = p + t n where p is the point and n is the normal of the plane.
• To reflect a point in a plane, find the equation of the line l between the point and the reflected point and substitute the i, j and k components into the x, y and z components of the cartesian plane equation. Then solve for the specific scalar parameter and substitute twice this into l to find the position.
• To reflect a line in a plane, find the intersection of the line and the plane. Then reflect any point on the line in the plane. Form an equation of the line between the reflected point and the intersection.
• The cross product of vectors a and b ( |a x b| ) is:

  | i j k |

  |a1 a2 a3 |

  |b1 b2 b3 |

  (a2b3 - a3b2) i + (a1b3 - a3b1) j + (a1b2 - a2b1) k.
• The scalar triple product of vectors a, b and c ( |a.(b x c)| ) is:

  |a1 a2 a3|

  |b1 b2 b3|

  |c1 c2 c3|

  a1(b2c3 - b3c2) + a2(b1c3 - b3c1) + a3(b1c2 - b2c1).
• The area of a triangle OAB is 1/2 |a x b| (the cross product).
• The area of a tetrahedron (triangular pyramid) with vectors a, b and c as three sides all coming from the same point is 1/6 |a . (b x c)| (the scalar triple product).
• To find the normal of a plane from its direction vectors, cross product them.