Question

$\overrightarrow{a}=3\hat{i}-5\hat{j}$ and $\overrightarrow{b}=8\hat{i}+3\hat{j}$ are two vectors. $\overrightarrow{c}$ is a vector such that $\overrightarrow{c}=\overrightarrow{a}\times \overrightarrow{b}$, then evaluate $|\overrightarrow{a}|:|\overrightarrow{b}|:|\overrightarrow{c}|$ ?

Answer 1

Note:
$\hat{i}\times \hat{i}=\hat{j}\times \hat{j}=0$
$\hat{i}\times \hat{j}=-\hat{j}\times \hat{i}=\hat{k}$
$\overrightarrow{c}=\overrightarrow{a}\times \overrightarrow{b}=(3\hat{i}-5\hat{j})\times (8\hat{i}+3\hat{j})=9\hat{k}+40\hat{k}=49\hat{k}$
$|\overrightarrow{a}|=\sqrt{3^2+5^2}=\sqrt{34}$
$|\overrightarrow{b}|=\sqrt{8^2+3^2}=\sqrt{73}$
$|\overrightarrow{c}|=\sqrt{{49}^2}=49$
$\therefore |\overrightarrow{a}|:|\overrightarrow{b}|:|\overrightarrow{c}|=\sqrt{34}:\sqrt{73}:49$