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 \(\displaystyle 4\mathbf{i}-2\mathbf{j}+\mathbf{k}\):

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