Let vectors $\displaystyle \vec{a}=\left \langle a1,a2\right \rangle$  and $\displaystyle \vec{b}=\left \langle b1,b2\right \rangle$.

The formula for the dot product is:

$\displaystyle \vec{a}\cdot\vec{b} = a1\cdot b1+ a2\cdot b2$

Follow this formula and simplify.

$\displaystyle \left \langle 1,6\right \rangle\cdot\left \langle 1,6\right \rangle = (1)(1)+(6)(6)=37$

The problem $\displaystyle \left \langle 3,4,5\right \rangle\cdot\left \langle1,2,3 \right \rangle$ is in the form of a dot product.  The final answer must be an integer, and not in vector form.

Write the formula for the dot product.

$\displaystyle \left \langle a_1,a_2,a_3\right \rangle\cdot\left \langle b_1,b_2,b_3\right \rangle=a_1b_1+a_2b_2+a_3b_3$

Substitute the givens and solve.

$\displaystyle (3)(1)+(4)(2)+(5)(3)=3+8+15=26$

Calculate $\displaystyle 2\vec{a}$.

$\displaystyle \vec{a}= \left \langle 2,4,-3\right \rangle$

$\displaystyle 2\vec{a}= \left \langle 4,8,-6\right \rangle$

Find the magnitude.

$\displaystyle |2\vec{a}|=\sqrt{4^2+8^2+(-6)^2}=\sqrt{16+64+36}=\sqrt{116}=2\sqrt{29}$

In order for the particles to collide, their $\displaystyle \small x$ and $\displaystyle \small y$ coordinates must be equal simultaneously. In order to check if this happens, we can set the particles' $\displaystyle \small x$-coordinates and $\displaystyle \small y$-coordinates equal to each other.

Let's start with the $\displaystyle \small x$-coordinate: 

$\displaystyle \small \small \small 3t^{2}-t=6\rightarrow 3t^2-t-6=0$.

Using the quadratic formula

$\displaystyle \small \small t=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$, we get $\displaystyle \small \small t=\frac{1+\sqrt{73}}{6}\approx1.59$ 

the other root is negative, so it can be discarded since $\displaystyle \small t>0$.

Now let's do the $\displaystyle \small y$-coordinate: $\displaystyle \small \small t+1/t=4\rightarrow t^2+1=4t\rightarrow t^2-4t+1=0$. Use the quadratic formula again to solve for $\displaystyle \small t$, and you'll get $\displaystyle \small 2\pm\sqrt{3}$ (these roots are approximately $\displaystyle \small 0.27$ and $\displaystyle \small 3.73$). The particles never have the same $\displaystyle \small x$ and $\displaystyle \small y$ coordinate simultaneously, so they do not collide.

$\displaystyle \vec{a}=< 3,2,1>$

$\displaystyle \vec{b}=< 1,0,10>$

$\displaystyle \vec{a}\cdot\vec{b}$ is simply the dot product of these two vectors. Mathematically, this is calculated as follows.

$\displaystyle (3\cdot 1)+(2\cdot 0)+(1\cdot 10)$  $\displaystyle =13$

To find the dot product of $\displaystyle a=\left \langle 3,4,7\right \rangle$ and $\displaystyle b=\left \langle -1,5,6\right \rangle$, calculate the sum of the products of the vectors' corresponding components:

$\displaystyle \left \langle 3,4,7\right \rangle\times \left \langle -1,5,6\right \rangle$

$\displaystyle =(3\times -1)+ (4\times 5)+(7\times 6)$

$\displaystyle =-3+ 20+42$

$\displaystyle =-3+ 62$

$\displaystyle =59$

To find the dot product of $\displaystyle a=\left \langle 5,-5,2\right \rangle$ and $\displaystyle b=\left \langle 2,3,8\right \rangle$, calculate the sum of the products of the vectors' corresponding components:

$\displaystyle \left \langle 5,-5,2\right \rangle\times \left \langle 2,3,8\right \rangle$

$\displaystyle =(5\times 2)+ (-5\times 3)+(2\times 8)$

$\displaystyle =10+ (-15)+16$

$\displaystyle =10+ 1$

$\displaystyle =11$

To find the dot product of $\displaystyle a=\left \langle 9,-8,7\right \rangle$ and $\displaystyle b=\left \langle 4,-1,2\right \rangle$, calculate the sum of the products of the vectors' corresponding components:

$\displaystyle \left \langle 9,-8,7\right \rangle\times \left \langle 4,-1,2\right \rangle$

$\displaystyle =(9\times 4)+ (-8\times -1)+ (7\times 2)$

$\displaystyle =36+ 8+ 14$

$\displaystyle =36+ 22$

$\displaystyle =58$

The dot product of two vectors is the sum of the products of their composite elements. Given $\displaystyle a=\left \langle 2,5,-4\right \rangle$ and $\displaystyle b=\left \langle 1,7,5\right \rangle$, the dot product would therefore be:

$\displaystyle \left \langle 2,5,-4\right \rangle\times \left \langle 1,7,5\right \rangle$

$\displaystyle =(2\times 1)+(5\times 7)+(-4\times 5)$

$\displaystyle =2+35+(-20)$

$\displaystyle =37+(-20)$

$\displaystyle =17$

The dot product of two vectors is the sum of the products of their composite elements. Given $\displaystyle a=\left \langle 6,7,-3\right \rangle$ and $\displaystyle b=\left \langle -4,-5,10\right \rangle$, the dot product would therefore be:

$\displaystyle \left \langle 6,7,-3\right \rangle\times \left \langle -4,-5,10\right \rangle$

$\displaystyle =(6\times -4)+(7\times -5)+(-3\times 10)$

$\displaystyle =(-24)+(-35)+(-30)$

$\displaystyle =(-24)+(-65)$

$\displaystyle =-89$