Question

Determine the projection of w onto u

Answer 1

c. 6.2i - 4.2j

Answer 2

c. 6.2i - 4.2j

Explanation:

The vector projection when the angle θ not known can be calculated using the following property of the dot product:

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

Where the dot product of two vectors is given by:

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

And the magnitude of a vector is given by:

`||u||=√(u_1^2+u_2^2+...+u_n^2)`

Using the previous definition, let's calculate the projection of w onto u:

First let's calculate the dot product between w and u:

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

Now let's find the magnitude of u:

`||u||=√(9^2+(-6)^2) = 3 √(13)`

So:

`||u||^2=117`

Therefore:

`proj_uw=(u\cdot w)/(||u||^2) u=(81)/(117) \langle9,-6\rangle=\langle6.23,-4.14\rangle\approx6.2i-4.2j`