Question

If vector A = i +2 j + k, vector B =2i +3k and vector C = 2i +4k , then what is the
magnitude of vector A + B -C ?, (I ,j, and k are unit vectors)

Answer 1

⇝ Given :-

$\vec{ \text A} = \hat\text i + 2 \hat\text j + \hat \text k$

$\vec{\text B} = 2\hat\text i + 3\hat \text k$

$\vec{ \text C} = 2 \hat\text i +4 \hat \text k$

⇝ To Find :-

$\text{Magnitude of} \: \vec{\text A} + \vec{\text B} - \vec{\text C}$

⇝ Solution :-

★ We Know magnitude of a vector $\vec{\text P} = \text x\hat\text i + \text y \hat\text j +\text z \hat \text k$is Given by The Formula :

$\pink{\underbrace{\large\bf |P| = \sqrt{ { x}^{2} + { y}^{2} + {z}^{2} }} }$

Now,

❒ Calculating Value of $\bf{\vec{A} + \vec{B} - \vec{C}}$

$\vec{ \text A} + \vec{\text B} - \vec{\text C}$

$= \hat\text i + 2 \hat\text j + \hat \text k + 2\hat\text i + 3\hat \text k - (2\hat\text i +4 \hat \text k)$

$=3 \hat\text i + 2 \hat{ \text j}+ \cancel{4\hat \text k } - 2\hat\text i - \cancel{4 \hat \text k}$

$\large= \hat{\text i} + 2 \hat{\text j}$

Hence,

$\large\underline{\pink{\underline{\frak{\pmb{ \vec{ \text{A}} + \vec{B} - \vec{ C} = \hat{i} + 2 \hat{j} }}}}}$

❒ Calculating Magnitude of $\bf{\vec{A} + \vec{B} - \vec{C}}$

$\sf |\vec{A} + \vec{B} - \vec{C}| = \sqrt{ {1}^{2} + {2}^{2} }$

$= \sqrt{1 + 4}$

$= \sqrt{5}$

Hence,

$\Large\underline{\pink{\underline{\frak{\pmb{|\vec{\text A} + \vec{B} - \vec{C}| = \sqrt{5} }}}}}$

Answer 2

Given :-

If vector  A = i +2 j + k, vector B =2i +3k and vector C = 2i +4k

To Find :-

Magnitude of vector A + B -C

Solution :-

$\sf A + B - C$

We have

A = i + 2j + k

B = 2i + 3k

C = 2i + 4k

(i + 2j + k) + (2i + 3k) - (2i + 4k)

i + 2j + k + 2i + 3k - 2i - 4k

i + 2i - 2i + k + 3k - 4k + 2j

3i - 2i + 4k - 4k + 2j

i + 2j

Now

Magnitude = √(i² + 2j²)

Magnitude = √(1² + 2²)

Magnitude = √(1 + 4)

Magnitude = √(5)

Magnitude = √5

${\frak{\pink{\underline{Hence,\;Magnitude\;is\;\sqrt{5}\;}}}}$