Question

Determine the projection of vector w onto vector u. u=9i-6j, v=-3i-2j, w=19i+15j

Answer 1

A on edge

Explanation:

Got it right on quiz (:

Answer 2

`proj_uw=6.2i-4.2j`

Explanation:

The projection of a vector
`v` onto a vector
`u` is defined as the projection of the vector
`v` on the line that contains the vector
`u`. It can be calculated using the following formula:

`proj_uv=(u\cdot v)/(||u||^2) u`

Where:

`u\cdot v`

Is the dot product between
`u` and
`v` which is given by:

`u\cdot v= $$\sum_(i=1)^(n) u_iv_i= u_1v_1+u_2v_2+...+u_nv_n$$`

and:

`||u||`

Is the magnitude of vector which can be calculated as follows:

`||u||=√(u_1^2+u_2^2+...+u_n^2)`

In this sense, the projection of vector w onto vector u is:

`proj_uw=(u\cdot w)/(||u||^2) u`

Where the dot product between
`u` and
`w` is:

`u\cdot w =(9*19)+(-6*15)=171-90=81`

And the magnitude of
`u` is:

`||u||=√(9^2+(-6)^2) = 3 √(13)`

Thus:

`proj_uw=(u\cdot w)/(||u||^2) u=(81)/(117) \langle9,-6\rangle=\langle6.23,-4.15\rangle\approx6.2i-4.2j`