Question

if a vector=3i^ +2j^ and b vector = i^ -2j^+3k^ . find the lenght a vector +b vector and a vector - b vector

Answer 1

Consider a vector:
$\vec{r} = a\hat{\imath} + b\hat{\jmath} + c\hat{k}$

The length of the vector, also called its modulus, is given as:


$| \vec{r} | = \sqrt{a^2+b^2+c^2}$

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Here, we have two vectors:

$\vec{a} = 3\hat{\imath}+2\hat{\jmath} \\ \\ \vec{b} = \hat{\imath}-2\hat{\jmath}+3\hat{k}$

We want to find the lengths of $\vec{a}+\vec{b}$ and $\vec{a}-\vec{b}$

Consider:

$\vec{c} = \vec{a}+\vec{b} \\ \\ \implies \vec{c} = (3\hat{\imath}+2\hat{\jmath})+(\hat{\imath}-2\hat{\jmath}+3\hat{k}) \\ \\ \implies \vec{c} = 4\hat{\imath}+3\hat{k} \\ \\ \implies | \vec{c} | = \sqrt{4^2+0^2+3^2} \\ \\ \implies | \vec{c} | = \sqrt{16+9} = \sqrt{25} \\ \\ \implies | \vec{c} | = 5 \\ \\ \\ \implies \boxed{| \, \vec{a}+\vec{b} \, | =5}$


Similarly, Consider:

$\vec{d} = \vec{a}-\vec{b} \\ \\ \implies \vec{d} = (3\hat{\imath}+2\hat{\jmath})-(\hat{\imath}-2\hat{\jmath}+3\hat{k}) \\ \\ \implies \vec{d} = 2\hat{\imath}+4\hat{\jmath}-3\hat{k} \\ \\ \implies | \vec{d} | = \sqrt{2^2+4^2+(-3)^2} \\ \\ \implies | \vec{d} | = \sqrt{4+16+9} \\ \\ \implies | \vec{d} | = \sqrt{29} \\ \\ \\ \implies \boxed{| \, \vec{a}-\vec{b} \, | =\sqrt{29}}$


Thus, Final Answers are:


$\boxed{\begin{array}{ccc} | \, \vec{a}+\vec{b} \, | & = & 5 \\ \\ | \, \vec{a}-\vec{b} \, | & = & \sqrt{29}\end{array}}$