ANSWER
[tex]A.\text{ }5\text{ m, }323.13\text{ deg}[/tex]
EXPLANATION
First, let us make a sketch of the vectors:
The thick black line represents the vector sum of the two vectors A and B.
We see that the coordinates of the vector (A + B) are:
[tex](A+B)=(4,-3)[/tex]
We can write it in component form as:
[tex]A+B=4i-3j[/tex]
To find the magnitude of (A + B), we have to find the length of the line using the formula:
[tex]L=|A+B|=\sqrt{x^2+y^2}[/tex]
where (x, y) are the coordinates of the vector.
Hence, the magnitude of the vector is:
[tex]\begin{gathered} |A+B|=\sqrt{(4)^2+(-3)^2}=\sqrt{16+9} \\ \\ |A+B|=\sqrt{25} \\ \\ |A+B|=5\text{ m} \end{gathered}[/tex]
To find the direction of the vector (A + B), we have to apply the formula:
[tex]\theta=\tan^{-1}(\frac{y}{x})[/tex]
Therefore, the direction of the vector is:
[tex]\begin{gathered} \theta=\tan^{-1}(-\frac{3}{4}) \\ \\ \theta=143.13\text{ deg or }323.13\text{ deg} \end{gathered}[/tex]
Since the vector is in the fourth quadrant, then, the direction is:
[tex]\theta=323.13\text{ deg}[/tex]
Therefore, the magnitude and direction of the sum vector (A + B) are:
[tex]A.\text{ }5\text{ m, }323.13\text{ deg}[/tex]
