What is the projection of << 3, -6, 2 >> onto << 3, 2, 1 >>?

2 Answers
May 31, 2016

It is <-3/14, -1/7, -1/14>.

Explanation:

The length of the projection of a vector v_1 on the vector v_2 is given by the dot product of v_1*v_2.
The dot product is the sum of the products of the components:

v_1=<3, -6, 2>
v_2=<3,2,1>

v_1*v_2=3*3+(-6)*2+2*1=9-12+2=-1.

The direction of the vector is the one of v_2 but we need to take the unitary vector in that direction that is the vector v_2 divided by its length.
The length is calculated again with the scalar product, this time of v_2 by itself (it is like to project the vector on itself)

||v_2||^2=v_2*v_2=3*3+2*2+1*1=14

So the unitary vector in the direction of v_2 is

\hat{v}_2=v_2/14=<3/14, 2/14, 1/14>=<3/14, 1/7, 1/14>

This unitary vector gives the direction, while the length and direction was given by the initial dot product that was -1.
So the final projection is

-1*\hat{v}_2=<-3/14, -1/7, -1/14>.

May 31, 2016

I got << -3/14,-1/7,-1/14 >>, and Wolfram Alpha confirms it.


PROJECTION OF \mathbf(veca) ONTO \mathbf(vecb)

Let veca = << 3,-6,2 >> and vecb = << 3,2,1 >>.

The projection of veca onto vecb, which yields a vector, is:

\mathbf("proj"_(vecb) veca = (vecacdotvecb)/(||vecb||cdot||vecb||)vecb)

This is drawn as:

For this, we should define a few more things.

  • The dot product of two n-long vectors vecu and vecv:

\mathbf(vecucdotvecv)

= << u_1,u_2, . . . ,u_N >> cdot << v_1,v_2, . . . ,v_N >>

= \mathbf(u_1v_1 + u_2v_2 + . . . + u_Nv_n)

  • The norm of vecv:

\mathbf(||vecv||)

= sqrt(vecvcdotvecv)

= sqrt(v_1v_1 + v_2v_2 + . . . + v_Nv_N)

= \mathbf(sqrt(v_1^2 + v_2^2 + . . . + v_N^2))

COMPONENTS OF THE PROJECTION

So, what we now have to do is:

color(green)(vecacdotvecb)

= << 3,-6,2 >>cdot<< 3,2,1 >>

= 3*3 + (-6*2) + 2*1

= 9 - 12 + 2

= color(green)(-1)

color(green)(||vecb||)

= sqrt(<< 3,2,1 >>cdot << 3,2,1 >>)

= sqrt(3*3 + 2*2 + 1*1)

= sqrt(9 + 4 + 1)

= color(green)(sqrt14)

FINAL CALCULATION

Finally, put it all back in the first equation.

color(blue)("proj"_(vecb) veca) = (vecacdotvecb)/(||vecb||cdot||vecb||)vecb

= (-1)/(sqrt14*sqrt14)<< 3,2,1 >>

= color(blue)(<< -3/14,-1/7,-1/14 >>)