First slide

Algebra of vectors

Question

if $\vec{r} and \vec{s}$ are non-zero constant vectors and the scalar D is chosen such that $| \vec{r} + b \vec{s} |$ is minimum, then the value of $| b \vec{s} |^{2} + | \vec{r} + b \vec{s} |^{2} =$

Moderate

A

$2 | \vec{r} |^{2}$
B

$\frac{| \vec{r} |^{2}}{2}$
C

$| \vec{r} |^{2}$
D

$3 | \vec{r} |^{2}$

Solution

For minimum value $| \vec{r} + b \vec{s} | = 0$
let $\vec{r} and \vec{s}$ are anti-parallel so $b \vec{s} = - \vec{r}$
$∴ | b \vec{s} |^{2} + | \vec{r} + b \vec{s} |^{2} = | - \vec{r} |^{2} + | \vec{r} - \vec{r} |^{2} = | \vec{r} |^{2}$

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