Vector Products

Question 1

what is the formula for the cross product of two vectors a and b

Answer 1

a x b = |a||b|sinθ*n

Question 2

what does i x j and j x i equal and why

Answer 2

• i x j = k
• j x i = -k
• because of the direction of sinθ (ACW is +ve)

Question 3

what do i x i, j x j and k x k equal

Answer 3

0

Question 4

what symbols signifies k being into the page

Answer 4

a circle with a cross in it

Question 5

what symbol signifies k being out of the page

Answer 5

a circle with a dot in it

Question 6

how do work out the unit vector of a vector

Answer 6

divide the vector by its modulus

Question 7

what is the formula for working out the volume of a parallelopiped with base vector lengths b and c, and vertical height a

Answer 7

V = a . (b x c)

Question 8

how could you modify the formula in a cyclic order

Answer 8

V = c . (a x b), you move the values to the right

Question 9

how could you modify the formula in an anti-cyclic order

Answer 9

V = -a . (c x b), the bracket values are swapped and there is a -ve sign

Question 10

what does the vector triple product generally look like

Answer 10

a x (b x c)

Question 11

what is the formula for working out a x (b x c)

Answer 11

a x (b x c) = b(a . c) - c(a . b)

Question 12

how would you remember the triple vector product formula

Answer 12

• 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

Question 13

how would you approach questions such as simplifying (a x b) . (a x c) x (a x d)

Answer 13

make (a x b) = e or something then use the formula

Question 14

does det(A) = det(A^T)

Answer 14

yep

Question 15

how could you rewrite (Ya) . (b x c) for simplifications sake (Y = lambda)

Answer 15

Y(a . b x c)

Question 16

what is the formula for the shortest distance between 2 lines (with line equations) and what are the variables

Answer 16

• 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)

Question 17

what is the formula for finding the shortest distance between a line and a point in 3D and what are the variables

Answer 17

• 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

Question 18

What is the formula for the shortest distance between a point and a plane and what are the variables

Answer 18

• D = |(b.n) - p| / |n|
• b = position vector of point
• n and p are from the equation of a plane r.n = p

Question 19

what is the formula for finding the shortest distance between a line ax + by = c and a point in 2D (x1, y1)

Answer 19

D = |ax1 + by1 - c| / sqrt(a^2 + b^2)