Let v and w Find the decomposition with respect to w Hint

Let v = and w =. Find the decomposition with respect to w. (Hint: is the vector projection of v onto w).

vk=dot product of v on w

Let i,j,k be the unit vectors along axises.

Then,

v=1i-j+3k

w=4i-2j+k

vk=(1.4+(-1.-2)+3.1)

vk=9

And v perpendicular is cross product of v on w i.e
v perp.= (1i-j+3k)*(4i-2j+k)

=i(-1+6)-j(1-12)+k(-2+4)

=5i+11j+2k

Decomposition=vk+v perp.

=9+5i+11j+2k