Vectors - Cross Product

Question

I was given the vector $v= <1 , 2 , -1>$ and was then asked to calculate $v \times i,\; v \times j,$ and $v \times k$. I know I would use the cross product for this, but is there a relationship between the original vector and $i,j,k $that I am missing?

Answer 1

$\vec v = i+2j-k$

$\vec v \times i = \begin{vmatrix}i&&j&&k\\1&&2&&-1\\1&&0&&0\end{vmatrix} = j-2k$

I assume you can do the others

EDIT:

You should think of $i= [1,0,0],j= [0,1,0]\& \;k = [0,0,1]$. Those are the basic definitions of the unit basis vectors. See here

Answer 2

Vector v can be represented as 1i +2j - 1k

For cross pdt with i you may proceed components wise,

1. Cross pdt of i and i (i X i)will result in 0

2.Cross pdt of of 2j and i (2j X i) will result in -2k

3. Cross pdt of -k and i (-k X i) will result in aj where a is magnitude i-e (-1 * 1)

Represent the results in steps 1,2,3 as a single vector to get the answer

For the possitve and negative sign of the direction vector(as in case 2, the resultant is negative contrary to the other two cases) you may imagine i to be in place of 12 in the clock, j in place of 4 and k in place of 8.

When the first vector is ai and second is of form bj, for moving from i to j you travel clockwise so the result (a*b)k will be possitive (because you moved clock wise)

If the first vector is j and second is i the resultant vector from the cross pdt. will be in direction of negative z axis as you travel anti clockwise while moving from j to i