Since vector B was not specified, I'll assume one at random. You can later answer your own question.
Answer:
[tex]\mid\mid \vec A+\vec B \mid \mid=\sqrt{105}[/tex]
[tex]\mid\mid \vec A-\vec B \mid \mid=\sqrt{149}[/tex]
Explanation:
Given:
[tex]\vec A=3\hat i +2\hat j+3\hat k[/tex]
And (assumed):
[tex]\vec B=-5\hat i +8\hat j-4\hat k[/tex]
Find the magnitude of
[tex]\vec A+\vec B[/tex]
[tex]\vec A-\vec B[/tex]
Given a vector
[tex]\vec P=x\hat i +y\hat j+z\hat k[/tex]
The magnitude of the vector is:
[tex]\mid\mid \vec P\mid \mid=\sqrt{x^2+y^2+z^2}[/tex]
[tex]\vec A+\vec B =3\hat i +2\hat j+3\hat k-5\hat i +8\hat j-4\hat k[/tex]
[tex]\vec A+\vec B =-2\hat i +10\hat j-\hat k[/tex]
The magnitude of the sum is:
[tex]\mid\mid \vec A+\vec B \mid \mid=\sqrt{(-2)^2+10^2+(-1)^2}=\sqrt{4+100+1}[/tex]
[tex]\mathbf{\mid\mid \vec A+\vec B \mid \mid=\sqrt{105}}[/tex]
[tex]\vec A-\vec B =3\hat i +2\hat j+3\hat k-(-5\hat i +8\hat j-4\hat k)[/tex]
[tex]\vec A-\vec B =3\hat i +2\hat j+3\hat k+5\hat i -8\hat j+4\hat k[/tex]
[tex]\vec A-\vec B =8\hat i -6\hat j+7\hat k[/tex]
The magnitude of the difference is:
[tex]\mid\mid \vec A-\vec B \mid \mid=\sqrt{8^2+(-6)^2+7^2}=\sqrt{64+36+49}[/tex]
[tex]\mathbf{\mid\mid \vec A-\vec B \mid \mid=\sqrt{149}}[/tex]
