# Two vectors are given by a = 3.3 x - 6.4 y and b = -17.8 x + 5.1 y. What is the angle between vector b and the positive x-axis?

Jul 8, 2017

$\phi = {164}^{\text{o}}$

#### Explanation:

Here's a more rigorous way to do this (easier way at the bottom):

We're asked to find the angle between vector $\vec{b}$ and the positive $x$-axis.

We'll imagine there is a vector that points in the positive $x$-axis direction, with magnitude $1$ for simplifications. This unit vector, which we'll call vector $\vec{i}$, would be, two dimensionally,

$\vec{i} = 1 \hat{i} + 0 \hat{j}$

The dot product of these two vectors is given by

vecb • veci = bicosphi

where

• $b$ is the magnitude of $\vec{b}$

• $i$ is the magnitude of $\vec{i}$

• $\phi$ is the angle between the vectors, which is what we're trying to find.

We can rearrange this equation to solve for the angle, $\phi$:

phi = arccos((vecb • veci)/(bi))

We therefore need to find the dot product and the magnitudes of both vectors.

The dot product is

vecb • veci = b_x i_x + b_yi_y = (-17.8)(1) + (5.1)(0) = color(red)(-17.8

The magnitude of each vector is

$b = \sqrt{{\left({b}_{x}\right)}^{2} + {\left({b}_{y}\right)}^{2}} = \sqrt{{\left(- 17.8\right)}^{2} + {\left(5.1\right)}^{2}} = 18.5$

$i = \sqrt{{\left({i}_{x}\right)}^{2} + {\left({i}_{y}\right)}^{2}} = \sqrt{{\left(1\right)}^{2} + {\left(0\right)}^{2}} = 1$

Thus, the angle between the vectors is

phi = arccos((-17.8)/((18.5)(1))) = color(blue)(164^"o"

Here's an easier way to do this:

This method can be used since we're asked to find the angle between a vector and the positive $x$-axis, which is where we typically measure angles from anyway.

Therefore, we can simply take the inverse tangent of vector $\vec{b}$ to find the angle measured anticlockwise from the positive $x$-axis:

$\phi = \arctan \left(\frac{5.1}{- 17.8}\right) = - {16.0}^{\text{o}}$

We must add ${180}^{\text{o}}$ to this angle due to the calculator error; $\vec{b}$ is actually in the second quadrant:

$- {16.0}^{\text{o" + 180^"o" = color(blue)(164^"o}}$