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i and j are unit vectors along x- and y- axis respectively. What is the magnitude and direction of the vectors

Q 4. 22  \hat i and \hat j are unit vectors along x- and y- axis respectively. What is the magnitude and direction of the vectors \hat i + \hat j, and \hat i - \hat j−? What are the components of a vectorA=2\hat i +3 \hat j along the directions of \hat i + \hat j and \hat i - \hat j? [You may use graphical method]

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Let A be a vector such that:-                 \overrightarrow{A}\ =\ \widehat{i}\ +\ \widehat{j}

Then the magnitude of vector A is given by   :            \left | A \right |\ =\ \sqrt{1^2\ +\ 1^2}\ =\ \sqrt{2}

Now let us assume that the angle made between vector A and x-axis is \Theta.

Then we have:-                 

                                                                \Theta \ =\ \tan^{-1}\left ( \frac{1}{1} \right )\ =\ 45^{\circ}

Similarly, let B be a vector such that:-       \overrightarrow{B}\ =\ \widehat{i}\ -\ \widehat{j}

The magnitude of vector B is     :                       \left | B\right |\ =\ \sqrt{1^2\ +\ (-1)^2}\ =\ \sqrt{2}

Let \alpha be the angle between vector B and x-axis :       

                                                                 \alpha \ =\ \tan^{-1}\left ( \frac{-1}{1} \right )\ =\ -45^{\circ}

 

Now consider   \overrightarrow{C}\ =\ 2\widehat{i}\ +\ 3\widehat{j} :-
Then the required components of a vector C along the directions of (\hat{i}+\hat{j})    is   :-     =\ \frac{2+3}{\sqrt{2}}\ =\ \frac{5}{\sqrt{2}}
and the required components of a vector C along the directions of (\hat{i}-\hat{j})   is    :-    \frac{2-3}{\sqrt{2}}\ =\ \frac{-1}{\sqrt{2}}

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