1
$\begingroup$

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?

$\endgroup$
2
  • $\begingroup$ 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
  • 1
    $\begingroup$ Sorry Gerry, I did more editing before I saw your comment. $\endgroup$
    – Erik M
    Mar 21 '14 at 22:09
0
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.