Find the point of intersection of the lines r(t) = (1, 0, 0) + t(−3, 5, 0) and s(t) = (0, 1, 1) + t(−2, 0, 5).

The point of intersection is:

Step-by-step explanation:

Let us combine the vectors to get each equation in the format of a single vector, and we should use a different parameter for the second line, I will use k:

Then we set the x,y and z components of the equations, equal to each other:

We have to solve that system of equations:

Solving the second and last for t and k we get:

We plug them into the first equation and we get:

Once we simplify:

So, the system actually has those solutions we have found for t and k. We can now use any of the equations of the two lines. Plugging   into the equation of the first line we get:

Therefore the point of intersection is:

use the power rule  

( a b ) n = a n b n

to distribute the exponent.

apply the product rule to  

3 x 2 y 3 . ( 3 x 2 ) 2 ( y 3 ) 2

apply the product rule to  

3 x 2 . 3 2 ( x 2 ) 2 ( y 3 ) 2

raise  

3

to the power of  

2 . 9 ( x 2 ) 2 ( y 3 ) 2 multiply the exponents in   ( x 2 ) 2

.

apply the power rule and multiply exponents,  

( a m ) n = a m n . 9 x 2 ⋅ 2 ( y 3 ) 2

multiply  

2   by   2 . 9 x 4 ( y 3 ) 2

multiply the exponents in  

( y 3 ) 2 .

apply the power rule and multiply exponents,  

( a m ) n = a m n . 9 x 4 y 3

⋅

2

multiply  

3   by   2 . 9 x 4 y 6

step-by-step explanation: