Answer:
(B) The magnitude is 7sqrt2, and the direction angle is approximately 135 degrees.
Step-by-step explanation:
From the figure, the coordinate of the tail of the vector,
[tex](x_1,y_1)=(3,-6).[/tex]
The coordinate of the head of the vector,
[tex](x_2,y_2)=(-4,1).[/tex]
The magnitude of a vector is the length between the head and tail of the vector.
By using the distance formula,
the magnitude of the vector [tex]= \sqrt{(1-(-6))^2+(-4-3)^2}[/tex]
[tex]=\sqrt{(7^2+(-7)^2}\\\\=\sqrt{98}\\\\=7\sqrt{2}[/tex]
Now, let [tex]\theta[/tex] be the angle made by the vector with the positive direction of the x-axis, so
[tex]\tan\theta = \frac{y_2-y_1}{x_2-x_1}\\\\\Rightarrow \tan\theta = \frac{1-(-6)}{-4-3}=\frac{7}{-7}=-1 \\\\\Rightarrow \theta = \tan^{-1}(-1) \\\\\Rightarrow \theta =\pi- \tan^{-1}1 \\\\\Rightarrow \theta = \pi-\frac{\pi}{4} \\\\\Rightarrow \theta =\frac{3\pi}{4} \\\\\Rightarrow \theta = 135 ^{\circ}.[/tex]
So, the magnitude of the vector is [tex]7\sqrt 2[/tex] which makes [tex]135 ^{\circ}[/tex] with the positive direction of the x-axis.
Hence, option (B) is correct.
