Question

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

Answer 1

Answer:

$proj_uw=6.2i-4.2j$

Step-by-step 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:

$proj_uv=\frac{u\cdot v}{||u||^2} u$

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:

$||u||=\sqrt{u_1^2+u_2^2+...+u_n^2}$

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

$proj_uw=\frac{u\cdot w}{||u||^2} u$

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:

$||u||=\sqrt{9^2+(-6)^2} = 3 \sqrt{13}$

Thus:

$proj_uw=\frac{u\cdot w}{||u||^2} u=\frac{81}{117} \langle9,-6\rangle=\langle6.23,-4.15\rangle\approx6.2i-4.2j$