The sum of two vectors 

\(\displaystyle a=< a_1, a_2>\)

\(\displaystyle b=< b_1, b_2>\)

is defined as 

\(\displaystyle a+b=< a_1+b_1, a_2+b_2>\)

For the vectors in this problem

\(\displaystyle u+v=< 4+2, 16+4>\)

\(\displaystyle =< 6,20>\)

To find the sum \(\displaystyle \begin{bmatrix} a&b \\ c&d \end{bmatrix}+\begin{bmatrix} e& f\\ g &h \end{bmatrix}=\begin{bmatrix} a+e& b+f\\ c+g&d+h \end{bmatrix}\). Using the matrices in the problem statement, we get  \(\displaystyle \begin{bmatrix} 1+3& 3+9\\ 4+17 &0+38 \end{bmatrix}\)=\(\displaystyle \begin{bmatrix} 4&12 \\ 21 &38 \end{bmatrix}\)

When adding vectors, we simply add the corresponding components together:

\(\displaystyle \left \langle a, b\right \rangle + \left \langle c, d\right \rangle= \left \langle a+c, b+d\right \rangle\)

Our answer is

\(\displaystyle \left \langle 10+x, x\right \rangle\)

Note that our final answer is a vector.

The sum of two vectors is the vector given by the sums of the respective components (for example, \(\displaystyle \left \langle a, b\right \rangle+\left \langle c, d\right \rangle=\left \langle a+c, b+d\right \rangle\)).

Our final answer is

\(\displaystyle \left \langle e^z(z^2+z), \cos(x)+\sin(x)\right \rangle\)

The formula for adding two vectors \(\displaystyle a=\left \langle x_1,y_1,z_1\right \rangle\) and \(\displaystyle b=\left \langle x_2,y_2,z_2\right \rangle\) is \(\displaystyle a+b=\left \langle x_1+x_2,y_1+y_2,z_1+z_2\right \rangle\). Using the vectors from the problem statement, this becomes \(\displaystyle \left \langle 2xy+7xy,3y+9y,z^3+2z^3\right \rangle=\left \langle 9xy,12y,3z^3\right \rangle\)

To add two vectors, we simply add the corresponding components (for example, \(\displaystyle \left \langle a, b\right \rangle+ \left \langle c, d\right \rangle=\left \langle a+c, b+d\right \rangle\))

Our final answer is

\(\displaystyle \left \langle 4\cos(yz), e^z+2e^x\right \rangle\)

To find the sum of two vectors \(\displaystyle a=\left \langle x_1,y_1,z_1\right \rangle\) and \(\displaystyle b=\left \langle x_2,y_2,z_2\right \rangle\) is \(\displaystyle a+b=\left \langle x_1+x_2,y_1+y_2,z_1+z_2\right \rangle\). Using the vectors we were given, we get \(\displaystyle a+b=\left \langle 3+5,y^2+5y^2,\sin(z)+3\sin(z)\right \rangle=\left \langle 8,6y^2,4\sin(z)\right \rangle\)

To find the sum of two vectors \(\displaystyle a=\left \langle x_1,y_1,z_1\right \rangle\) and \(\displaystyle b=\left \langle x_2,y_2,z_2\right \rangle\) is \(\displaystyle a+b=\left \langle x_1+x_2,y_1+y_2,z_1+z_2\right \rangle\). Using the vectors we were given, we get \(\displaystyle a+b=\left \langle 2+10,y^4+3y^4,xyz+2xyz\right \rangle=\left \langle 12,4y^4,3xyz\right \rangle\)

To find the sum of two vectors \(\displaystyle a=\left \langle x_1,y_1,z_1\right \rangle\) and \(\displaystyle b=\left \langle x_2,y_2,z_2\right \rangle\) is \(\displaystyle a+b=\left \langle x_1+x_2,y_1+y_2,z_1+z_2\right \rangle\). Using the vectors we were given, we get \(\displaystyle a+b=\left \langle -4+2,3y^3+y^3,z+5z\right \rangle=\left \langle -2,4y^3,6z\right \rangle\)

To find the sum of two vectors \(\displaystyle a=\left \langle x_1,y_1,z_1\right \rangle\) and \(\displaystyle b=\left \langle x_2,y_2,z_2\right \rangle\) is \(\displaystyle a+b=\left \langle x_1+x_2,y_1+y_2,z_1+z_2\right \rangle\). Using the vectors we were given, we get \(\displaystyle a+b=\left \langle 5-4,3xyz+2xyz,1-5\right \rangle=\left \langle 1,5xyz,-4\right \rangle\)