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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 the x - and y -axis, respectively. What is the magnitude and direction of the vectors $\hat{i}+\hat{j}$, and $\hat{i}-\hat{j}_{\text {? What are the components of a vector }} A=2 \hat{i}+3 \hat{j}$ along the directions of $\hat{i}+\hat{j}$ and $\hat{i}-\hat{j}_{\text {? [You may use the graphical }}$ method]

Answers (1)

best_answer

Let A be a vector such that:-

$$
\vec{A}=\widehat{i}+\widehat{j}
$$


Then the magnitude of vector A is given by :

$$
|A|=\sqrt{1^2+1^2}=\sqrt{2}
$$


Now let us assume that the angle made between vector $A$ and the $x$-axis is $\Theta$.
Then we have:-

$$
\begin{aligned}
\Theta & =\tan ^{-1}\left(\frac{1}{1}\right)=45^{\circ} \\
\vec{B} & =\widehat{i}-\widehat{j}
\end{aligned}
$$


Similarly, let B be a vector such that:-

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


The magnitude of vector $B$ is :

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

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


Now consider $\vec{C}=2 \widehat{i}+3 \widehat{j}$ :-
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:- $\frac{2-3}{\sqrt{2}}=\frac{-1}{\sqrt{2}}$

Posted by

Devendra Khairwa

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