Projection of u onto v and v onto u

Given the vector$u=<-2,6,4>$ and a vector v such that the vector projection of u onto v is $<2,4,4>$ , and the vector projection of v onto u is $<-8,24,16>$ . What is the vector v?

Let$\overrightarrow{v}=<a,b,c>$

Projection of$\overrightarrow{v}\text{}on\to \text{}\overrightarrow{u}$ is given by:

$pro{j}_{u}v=\frac{\overrightarrow{u}.\overrightarrow{v}}{{\left|u\right|}^{2}}\overrightarrow{u}=\frac{\overrightarrow{u}.\overrightarrow{v}}{{(-2)}^{2}+{6}^{2}+{4}^{2}}<-2,6,4>$

$<-8,24,16\ge \frac{\overrightarrow{u}.\overrightarrow{v}}{{(-2)}^{2}+{6}^{2}+{4}^{2}}<-2,6,4>$

$\Rightarrow 4<-2,6,4\ge \frac{\overrightarrow{u}.\overrightarrow{v}}{4+36+16}<-2,6,4>$

$\Rightarrow <-2,6,4\ge \frac{\overrightarrow{u}.\overrightarrow{v}}{224}<-2,6,4>$

On comparing$\frac{\overrightarrow{u}.\overrightarrow{v}}{224}=1$

$\Rightarrow \overrightarrow{u}.\overrightarrow{v}=224$

$pro{j}_{v}u=\frac{\overrightarrow{u}.\overrightarrow{v}}{{\left|v\right|}^{2}}\overrightarrow{v}=\frac{224}{{\left|v\right|}^{2}}<a,b,c>$

$\Rightarrow <2,4,4\ge \frac{224}{{\left|v\right|}^{2}}<a,b,c>$

Dividing both sides by 2 we get:

$\Rightarrow <1,2,2\ge \frac{112}{{\left|v\right|}^{2}}<a,b,c>$

Given the vector

Let

Projection of

On comparing

Dividing both sides by 2 we get: