Question

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Plz answer questions - 28 and 29

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Answer 1

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$\huge{\bold{\underline{Given:-}}}$

$\large{ \vec{A} = 3 \hat{i} + 2 \hat{j}}$

$\huge{\bold{\underline{Explanation:-}}}$

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28 th question:-

As we know,

$\large{\boxed{ Component \: of \: A = \dfrac{ \vec{A} . \vec{B}}{| \vec{B} |} \: ---(1)}}$

Now,

Solving separately,

Remember ${ \vec{B} = \hat{i} + \hat{j}}$

Now,

$\large{ \vec{A} . \vec{B}}$

(Note we are doing dot product)

Substituting the values,

$\large{ \vec{A} . \vec{B} = [3 \hat{i} + 2 \hat{j}] . [ \hat{i} + \hat{j}]}$

Taking dot product gives,

$\large{ \vec{A} . \vec{B} = 3(1) + 2(1)}$

$\large{ \vec{A} . \vec{B} = 5}$

Now,

Magnitude of $\large{ \vec{B}}$

${ \vec{B} = \hat{i} + \hat{j}}$

$\large{ | \vec{B}| = \sqrt{(1)^2 + (1)^2}}$

Simplifying,

$\large{ | \vec{B}| = \sqrt{2}}$

Substituting in equation (1).

$\large{ component = \dfrac{5}{ \sqrt{2}}}$

The above equation is in scalar form,

Making it to vector form ,

Before that finding the unit vector along ${ \vec{i} + \vec{j}}$

As we know,

$\large{ \hat{a} = \dfrac{ \vec{i} + \vec{j}}{| \vec{i} + \vec{j}|}}$

Simplifying,

$\large{ \hat{a} = \dfrac{\vec{i} + \vec{j}}{ \sqrt{2}}}$

Now, vector component,

$\large{ Component = \dfrac{5}{ \sqrt{2}} \times \dfrac{\vec{i} + \vec{j}}{ \sqrt{2}}}$

It becomes,

$\large{ component = \dfrac{5}{2} \times [\vec{i} + \vec{j}]}$

$\huge{\boxed{\boxed{ Component = \dfrac{5}{2} (\vec{i} + \vec{j})}}}$

Same procedure can be used for Finding the component

along the direction ${(\vec{i} - \vec{j})}$

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29th question:-

As given:-

$\large{ | \vec{A} + \vec{B}| = \sqrt{3}}$

And it is given that A and B are unit vectors.

I. e.

$\large{ | \vec{A}| = 1 \: and \: | \vec{B}| = 1}$

#refer the attachment for the answer,

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