Let $v_1$, $v_2$, $v_3$ be mutually orthogonal non-zero vectors in 3-space. So, any vector $v$ can be expressed as $v=c_1v_1+c_2v_2+c_3v_3$.

(a) Show that the scalars $c_1$, $c_2$, $c_3$ are given by the formula $$c_i=v\cdot \frac{v_i}{|v_i|^2},i=1,2,3$$

(b) Show that \begin{align*} v_1 &= 3i − j + 2k,\\ v_2 &= i + j − k,\\ v_3 &= i − 5j − 4k \end{align*} are mutually orthogonal. Now let $v = i − j + k$. Use the result from (a) to find scalars $c_1$, $c_2$ and $c_3$ such that $$v = i − j + k = c_1v_1 + c_2v_2 + c_3v_3$$

how to calculate the value of $v\cdot v_i$? what distributivity of scalar multiplication and addition?

  • $\begingroup$ I did a little formatting on a) --- maybe you could have a look at what I did, and do something similar for the rest of the question. $\endgroup$ Mar 21 '14 at 22:05
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    $\begingroup$ Sorry Gerry, I did more editing before I saw your comment. $\endgroup$
    – Erik M
    Mar 21 '14 at 22:09

To specifically answer your question, suppose that $\langle\cdot,\cdot\rangle$ is the Euclidean inner product (i.e. $\langle \vec{u},\vec{v}\rangle=\vec{u}\cdot \vec{v}=\vec{u}^T\vec{v}$).

You can show that the inner product is distributive over vector addition $$ \langle \vec{a},\vec{b}+\vec{c}\rangle=\langle \vec{a},\vec{b}\rangle+\langle \vec{a},\vec{c}\rangle $$ and scalar multiplication $$ \langle \beta \vec{a},\vec{b}\rangle=\beta\langle \vec{a},\vec{b}\rangle. $$

These properties need to be used to solve a.


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