Vector Products Flashcards
what is the formula for the cross product of two vectors a and b
a x b = |a||b|sinθ*n
what does i x j and j x i equal and why
- i x j = k
- j x i = -k
- because of the direction of sinθ (ACW is +ve)
what do i x i, j x j and k x k equal
0
what symbols signifies k being into the page
a circle with a cross in it
what symbol signifies k being out of the page
a circle with a dot in it
how do work out the unit vector of a vector
divide the vector by its modulus
what is the formula for working out the volume of a parallelopiped with base vector lengths b and c, and vertical height a
V = a . (b x c)
how could you modify the formula in a cyclic order
V = c . (a x b), you move the values to the right
how could you modify the formula in an anti-cyclic order
V = -a . (c x b), the bracket values are swapped and there is a -ve sign
what does the vector triple product generally look like
a x (b x c)
what is the formula for working out a x (b x c)
a x (b x c) = b(a . c) - c(a . b)
how would you remember the triple vector product formula
- the value outside the bracket is the first value in the 2 other brackets
- the first value in the bracket is on the outside of the first bracket
- the second value in the bracket is on the outside o the second bracket
- then the last place is filled with the missing value
how would you approach questions such as simplifying (a x b) . (a x c) x (a x d)
make (a x b) = e or something then use the formula
does det(A) = det(A^T)
yep
how could you rewrite (Ya) . (b x c) for simplifications sake (Y = lambda)
Y(a . b x c)
what is the formula for the shortest distance between 2 lines (with line equations) and what are the variables
- D = |(b-a) . n| / |n|
- b = position vector of line 2
- a = position vector of line 1
- n = vector perpendicular to both direction vectors (x product)
what is the formula for finding the shortest distance between a line and a point in 3D and what are the variables
- D = |(p - a) x d| / |d|
- a = the position vector of the line
- d = the direction vector of the line
- p = the given point
- x = cross product
What is the formula for the shortest distance between a point and a plane and what are the variables
- D = |(b.n) - p| / |n|
- b = position vector of point
- n and p are from the equation of a plane r.n = p
what is the formula for finding the shortest distance between a line ax + by = c and a point in 2D (x1, y1)
D = |ax1 + by1 - c| / sqrt(a^2 + b^2)
