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

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

To find the dot product between the 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 from the problem statement, we get

$\displaystyle (3x*x^2)+(y*y^3)+(5z*z^4)=3x^3+y^4+5z^5$

o find the dot product between the 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 from the problem statement, we get

$\displaystyle (-4*x)+(-10*y^2)+(1*z)=-4x-10y^2+z$

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

Our final answer is

$\displaystyle 2z+e^xy^3z^2+x^2$

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 apply the formula:

$\displaystyle a\cdot b=(x_1*x_2)+(y_1*y_2)+(z_1*z_2)$

Using the vectors from the problem statement, we get

$\displaystyle (x*3)+(y*-7)+(4z*1)=3x-7y+4z$

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 apply the formula:

$\displaystyle a\cdot b=(x_1*x_2)+(y_1*y_2)+(z_1*z_2)$

Using the vectors from the problem statement, we get

$\displaystyle (4*2)+(1*-8)+(-5*1)=-5$

To find the dot product between vectors $\displaystyle a=\left \langle x_1,y_1,z_1\right \rangle$ and $\displaystyle b=\left \langle x_2,y_2,z_2\right \rangle$, you apply the following formula:

$\displaystyle a\cdot b=(x_1*x_2)+(y_1*y_2)+(z_1*z_2)$. Applying to the vectors from the problem statement, we get:

$\displaystyle (3*1)+(0*-2)+(-6*3)=-15$

To find the dot product between vectors $\displaystyle a=\left \langle x_1,y_1,z_1\right \rangle$ and $\displaystyle b=\left \langle x_2,y_2,z_2\right \rangle$, you apply the following formula:

$\displaystyle a\cdot b=(x_1*x_2)+(y_1*y_2)+(z_1*z_2)$. Applying to the vectors from the problem statement, we get:

$\displaystyle (x^3*4x^3)+(3*-7)+(1*2)=4x^6-19$

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

Our final answer is

$\displaystyle 3xy+x^2yz^2+yz^3$