The formula for the dot product between 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 $\displaystyle a\cdot b=(x_1*x_2)+(y_1*y_2)+(z_1*z_2)$. Using the vectors from the problem statement, we get $\displaystyle (3*1)+(10*2)+(6*5)=53$

The dot product of two vectors is given by the sum of the products of the corresponding components (for example, $\displaystyle \left \langle a, b\right \rangle \cdot \left \langle x, y\right \rangle=ax+by$)

Our final answer is

$\displaystyle 3\sec(z)+14\tan(z)$

To find the dot product between two vectors $\displaystyle a=\left \langle x_1,y_1,z_1\right \rangle$ and $\displaystyle b=\left \langle x_2,y_2,z_2\right \rangle$ we use the formula $\displaystyle a\cdot b=(x_1*x_2)+(y_1*y_2)+(z_1*z_2)$. Using the vectors in the problem statement, we get $\displaystyle (1*2)+(-4*2)+(5*1)=-1$

To find the dot product between two vectors $\displaystyle a=\left \langle x_1,y_1,z_1\right \rangle$ and $\displaystyle b=\left \langle x_2,y_2,z_2\right \rangle$ we use the formula $\displaystyle a\cdot b=(x_1*x_2)+(y_1*y_2)+(z_1*z_2)$. Using the vectors in the problem statement, we get $\displaystyle (\sin(x)*\csc(x))+(y*2)+(5*4)=2y+21$

To find the dot product between two vectors $\displaystyle a=\left \langle x_1,y_1,z_1\right \rangle$ and $\displaystyle b=\left \langle x_2,y_2,z_2\right \rangle$ we use the formula $\displaystyle a\cdot b=(x_1*x_2)+(y_1*y_2)+(z_1*z_2)$. Using the vectors in the problem statement, we get $\displaystyle (x*2)+(2y*2)+(z*1)=2x+4y+z$

To find the dot product between two vectors $\displaystyle a=\left \langle x_1,y_1,z_1\right \rangle$ and $\displaystyle b=\left \langle x_2,y_2,z_2\right \rangle$ we use the formula $\displaystyle a\cdot b=(x_1*x_2)+(y_1*y_2)+(z_1*z_2)$. Using the vectors in the problem statement, we get $\displaystyle (3*3x)+(2y*yz)+(z^5*z^2)=9x+2y^2z+z^7$

To find the dot product between two vectors $\displaystyle a=\left \langle x_1,y_1,z_1\right \rangle$ and $\displaystyle b=\left \langle x_2,y_2,z_2\right \rangle$ we use the formula $\displaystyle a\cdot b=(x_1*x_2)+(y_1*y_2)+(z_1*z_2)$. Using the vectors in the problem statement, we get $\displaystyle (3x*4)+(4y*2)+(z*7)=12x+8y+7z$

To find the dot product between two vectors $\displaystyle a=\left \langle x_1,y_1,z_1\right \rangle$ and $\displaystyle b=\left \langle x_2,y_2,z_2\right \rangle$ we use the formula $\displaystyle a\cdot b=(x_1*x_2)+(y_1*y_2)+(z_1*z_2)$. Using the vectors in the problem statement, we get $\displaystyle (2*5)+(-3*-4)+(4*1)=10+12+4=26$

To find the dot product between two vectors $\displaystyle a=\left \langle x_1,y_1,z_1\right \rangle$ and $\displaystyle b=\left \langle x_2,y_2,z_2\right \rangle$ we use the formula $\displaystyle a\cdot b=(x_1*x_2)+(y_1*y_2)+(z_1*z_2)$. Using the vectors in the problem statement, we get $\displaystyle (x*x)+(-3*y)+(4z*z)=x^2-3y+4z^2$

Two vectors are defined as orthogonal when their dot product is zero.

The dot product of two vectors

 $\displaystyle \mathbf{u}=u_x\mathbf{i}+u_y\mathbf{j}+u_z\mathbf{k}$   and   $\displaystyle \mathbf{v}=v_x\mathbf{i}+v_y\mathbf{j}+v_z\mathbf{k}$

Is given by the expression:

$\displaystyle \mathbf{u}\cdot\mathbf{v}=u_xv_x+u_yv_y+u_zv_z$

The only vector that satisfies the requirement that the dot product of it and v is zero is $\displaystyle 4\mathbf{i}-2\mathbf{j}+\mathbf{k}$:

$\displaystyle (3\mathbf{i}+7\mathbf{j}+2\mathbf{k})\cdot (4\mathbf{i}-2\mathbf{j}+\mathbf{k}) =(3)(4)+(7)(-2)+(2)(2)=12-14+2=0$