Vectors

• A vector can be denoted by $\vec a$
• The vector equation of a line is in the form r = a + td, where a is a general point on the line and d is the direction vector of the line
• If two lines are parallel, one direction vector will be a multiple of the other
• If two direction vectors are parallel and there is a value of t that gives the general point on the other line when put into one equation, the two equations represent the same line
• To write the cartesian equation of a line from a vector equation, write x, y and z in terms of t and rearrange to make t the subject. Then set each equation equal to each other (unless the coefficient of t is 0, then write this separately). The final equation should be in the following form:

  $(x+a)/b =$$(y+c)/d =$$(z+e)/f$
• To find the intercepts of 2 lines, set each dimension in both equations equal to each other and solve the first 2 equations simultaneously, then check the third equation
• For skew lines, the direction vectors are not parallel, but there is not a set of solutions that works for every equation
• The dot product of two direction vectors is a1*b1 + a2*b2 + a3*b3 if the two vectors are in the form:

  a1 b1

  a2 and b2

  a3 b3
• Two vectors are perpendicular if $\vec a . \vec b =$$0$
• The cross product is the vector perpendicular to the other two vectors. It is given by the determinant of the 3x3 matrix formed by putting i, j and k in the first column and the other two vectors in the second and third
• $\vec a X \vec b$ = $- \vec b X \vec a$
• $cos(\theta)$ = $\frac {\vec a . \vec b} {|\vec a||\vec b|}$