Question

What is the angle made by 3 i ˆ + 4 j ˆ with x-axis ?

Answer 1

About `53.13` degrees.

Explanation:

The vector `3\hat{i}+4\hat{j}` identifies the point `(3,4)` when considering the bijection `phi: \mathbb{R}^2\to\mathbb{R}^2` defined by `phi(a\hat{i}+b\hat{j}) = (a,b)`.

In other words, given a vector, we put its beginning on the origin `(0,0)`, and identify the vector with its end.

But we know how to compute the angle with the `x`-axis for any given point `P=(x,y)` on the plain: we have

`\theta = arctan(y/x)`

So, you have

`\theta = arctan(4/3) = 53.13`