Answer:
The magnitude is 96.67 lb and the direction angle from the positive x-axis is 13.65º.
Step-by-step explanation:
To find the effect of combining the two forces, add their representative vectors.
First, express each vector in component form or in terms of the standard unit vectors. For this purpose, it is easiest if we align one of the vectors with the positive x-axis. The horizontal vector, then, has initial point (0, 0) and terminal point (45, 0). It can be expressed as
[tex]45\mathbf{i}[/tex]
The second vector has magnitude 54 and makes an angle of 25° with the first, so we can express it as
[tex]54\:\cos(25)\mathbf{i}+54\:\cos(25)\mathbf{j}[/tex]
Then, the sum of the vectors, or resultant vector, is
[tex]\mathbf{r}=(45+54\:\cos(25))\mathbf{i}+54\:\sin(25)\mathbf{j}[/tex]
The magnitude is
[tex]||\mathbf{r}||=\sqrt{(45+54\:\cos(25))^{2} +(54\:\sin(25)} )^{2}\\\\||\mathbf{r}||=9\sqrt{60\cos \left(25^{\circ \:}\right)+61}\\\\||\mathbf{r}||\approx96.67[/tex]
The angle [tex]\theta[/tex] made by [tex]\mathbf{r}[/tex] and the positive x-axis has
[tex]\tan \theta =\frac{54\:\sin \left(25\right)}{45+54\:\cos \left(25\right)} \approx 0.24293\\\\\theta=\tan^{-1} (0.24293)=13.65^{\circ \:}[/tex]
which means the resultant force [tex]\mathbf{r}[/tex] has an angle of 13.65° above the horizontal axis.
