Question

Write the vector u as a sum of two orthogonal vectors, one of which is the vector projection of u onto v, projvu u=(-8,8) , v= (-1,2)
A. (-0.5,1) + ( 7.5,-9) C. (-1,2) + (-2,-1)


B (-1,2) + (-7,-10) D (1.6,-3.2)+ (-9.6,-4.8)

Answer 1

Answer:

none of the given options is true

Step-by-step explanation:

Given: u=(-8,8) , v= (-1,2)

To find: vectors $w_1,w_2$ such that $u=w_1+w_2$

Solution:

A vector is a quantity that has both magnitude and direction.

$proj_vu=\frac{u\cdot v}{\left | v \right |^2}(v)\\=\frac{(-8,8)\cdot (-1,2)}{(\sqrt{ (-1)^2+2^2})^2}(-1,2)\\=\frac{8+16}{5}(-1,2)\\=\frac{1}{5}(-24,48)\\=(-4.8,9.6)$

Let $w_1=(-4.8,9.6)$

So,

$w_2=u-w_1\\=(-8,8)-(-4.8,9.6)\\=(-8+4.8,8-9.6)\\=(-3.2,-1.6)$

So, u = $(-4.8,9.6)+(-3.2,-1.6)$

So, none of the given options is true