Let L1 be the line passing through the points Q1=(−2, 4, −2) and Q2=(−3, 5, 1) and let L2 be the line passing through the point P1=(−2, 10, −5) with direction vector →d=[−2, 6, 4]T. Determine whether L1 and L2 intersect. If so, find the point of intersection Q.

The answers to the questions are;

Yes, the lines  L1 and L2 intersect.

The point of intersection Q of the lines L1 and L2 is (1, 1, -11).

Step-by-step explanation:

The direction vector for L1 is (1, -1, -3)

and an equation is

x = -3 + t

y =  5 - t

z = 1 - 3·t

The possible equation for L2 is

x = -2 - 2·k

y = 10 + 6·k

z = -5 + 4·k

For intersection, we have

-3 + t = -2 - 2·k AND 5 - t = 10 + 6·k AND 1 - 3·t = -5 + 4 k

From the first equation, we have

t = 1 - 2·k and substituting in the second equation, we get

5 - (1 - 2·k) = 10 + 6·k

4 + 2·k = 10 + 6·k

6 = -4·k and k = -3/2

From which t = 1 - 2·k = 1 -2×(-3/2) = 1 + 3 = 4

t = 4

We now check if the results for t and k satisfies the third equation as follows

Left Hand Side is  1 - 3·t = 1 - 3×4 = -11

Right Hand Side is  -5 + 4 k = -5 + (4×(-3/2)) = -5 -6 = -11

Therefore the results from the first two equations satisfy the third equation and we now look for the intersection point as

x = -3 + t  ⇒ -3 + 4 = 1

y =  5 - t ⇒    5 - 4 = 1

z = 1 - 3·t ⇒    1 - 3×4 = -11

Giving the intersection point Q as (1, 1, -11).

the entire expression should be subtracted from 30. to subtract the quantity, distribute the negative sign, which changes the signs on the terms. so, the –2 changes to positive and gets added to 30 instead of subtracted from it.

hope this ! ^w^

25.2

step-by-step explanation:

2+4x6-8/10= 25.2