Question

i and j are unit vectors along x-axis and y-axis respectively. What is the magnitude and direction of the vectors i + j and i - j? What are the components of a vector A = 2i + 3j along the directions of i + j and i - j? [You may use graphical method]

Answer 1

Let , $\vec{a}=\hat{i}+\hat{j}$
Magnitude of $\vec{a}=\sqrt{1^2+1^2}=\sqrt{2}$
Let $\theta$ is the angle made by $\vec{a}$ with x-axis
Then, value of $\theta=tan^{-1}\frac{1}{1}=tan^{-1}1=45^{\circ}$ , direction is shown in figure of $\vec{a}$

Similarly, Let $\vec{b}=\hat{i}-\hat{j}$
Magnitude of $\vec{b}=\sqrt{1^2+(-1/^2}=\sqrt{2}$
angle made by $\vec{b}$ with x-axis is $tan^{-1}\frac{-1}{1}=tan^{-1}(-1)=-45^{\circ}$, direction is shown in figure .

Now, Let $\vec{A}=2\hat{i}+3\hat{j}$
The components of a vector A= 2i + 3j along the directions of $(\hat{i}+\hat{j})$ : $\vec{A}(2,3)$
The components of a vector A = 2i + 3j along the directions of $(\hat{i}-\hat{j})$ : $\vec{A}(2,-3)$

Answer 2

here is the answer

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