I am rather new to calculus, and am trying to resolve the following question. I have come to an answer, but it is not listed amongst the possible answers, so I would love to know where my reasoning failed...
Vector $\mathbf u = (1,0,1)$ is split into two perpendicular vectors where one is in the direction of vector $\mathbf b = (1,1,2)$. Calculate the value of $\left\lVert \mathbf u \right\rVert ^2$$\left\lVert \mathbf b \right\rVert ^2$$\left\lVert \mathbf v \right\rVert ^2$$\left\lVert \mathbf w \right\rVert ^2$.
$\left\lVert \mathbf b \right\rVert=\sqrt6$
$\left\lVert \mathbf u \right\rVert=\sqrt2$
$cosθ = \frac {u·b}{\left\lVert \mathbf u \right\rVert \left\lVert \mathbf b \right\rVert }$=$\frac{(1,0,1)(1,1,2)}{\sqrt12}$=$\frac {\sqrt 3}{2}$, where θ refers to the angle between $\mathbf u$ and $\mathbf b$; therefore θ = 30 degrees.
Therefore $\left\lVert \mathbf v \right\rVert$ is equal to $\sqrt 2 · cos60^\circ$=$\frac {1}{2}\sqrt2$.
Therefore $\left\lVert \mathbf w \right\rVert$ is equal to $\sqrt 2 · sin60^\circ$=$\frac {\sqrt6}{2}$.
However, that leads to a result of 9 when put the norms into the equation mentioned above, and the options given are 1,2,3 and 4.
Can anybody please let me know where I went wrong?
Thank you!