Answer:
c. 6.2i - 4.2j
Step-by-step explanation:
The vector projection when the angle θ not known can be calculated using the following property of the dot product:
[tex]proj_uv=\frac{u\cdot v}{||u||^2} u[/tex]
Where the dot product of two vectors is given by:
[tex]u\cdot v= $\sum_{i=1}^{n} u_iv_i= u_1v_1+u_2v_2+...+u_nv_n$[/tex]
And the magnitude of a vector is given by:
[tex]||u||=\sqrt{u_1^2+u_2^2+...+u_n^2}[/tex]
Using the previous definition, let's calculate the projection of w onto u:
First let's calculate the dot product between w and u:
[tex]u\cdot w =(9*19)+(-6*15)=171-90=81[/tex]
Now let's find the magnitude of u:
[tex]||u||=\sqrt{9^2+(-6)^2} = 3 \sqrt{13}[/tex]
So:
[tex]||u||^2=117[/tex]
Therefore:
[tex]proj_uw=\frac{u\cdot w}{||u||^2} u=\frac{81}{117} \langle9,-6\rangle=\langle6.23,-4.14\rangle\approx6.2i-4.2j[/tex]
