To make it easier to refer to them, let's call the first vector →u and the second →v. We want the project of →u onto →v:
proj→v→u=⎛⎜
⎜⎝→u⋅→v∣∣∣∣→v∣∣∣∣2⎞⎟
⎟⎠⋅→v
That is, in words, the projection of vector →u onto vector →v is the dot product of the two vectors, divided by the square of the length of →v times vector →v. Note that the piece inside the parentheses is a scalar that tells us how far along the direction of →v the projection reaches.
First, let's find the length of →v:
∣∣∣∣→v∣∣∣∣=√32+22+(−3)2=√22
But note that in the expression what we actually want is ∣∣∣∣→v∣∣∣∣2, so if we square both sides we just get 22.
Now we need the dot product of →u and →v:
→u⋅→v=(1×3+(−2)×2+3×(−3))=(3−4−9)=(−10)
(to find the dot product we multiply the coefficients of i,jandk and add them)
Now we have everything we need:
proj→v→u=⎛⎜
⎜⎝→u⋅→v∣∣∣∣→v∣∣∣∣2⎞⎟
⎟⎠⋅→v=(−1022)(3i+2j−3k)
=(−3022i−2022j+3022k)=(−1511i−1011j+1511k)
