How do I show two vectors are perpendicular?

Vectors can describe a line of particular length ("magnitude") and direction. The angle x between two vectors a and b can be found using the formula a.b = |a| |b| cosx. For the vectors to be perpendicular (at right angles) then cosx = 0, so we know that the dot product or scalar product a.b must = 0. If you calculate the scalar product and show it = 0 the vectors must be perpendicular.
To calculate the scalar product of two vectors eg a = 3i + 4j - 12k and b= 4i + 3j + 2k we simply multiply the two i terms, the two j terms, the two k terms and add them all up, being careful with the + or - signs. So here a.b = 12 + 12 - 24 = 0. Therefore a and b are perpendicular.