Question 3 Determine for what values of the parameter a ? the following planes i

Question

Question 3

Determine for what values of the parameter a ? the following planes in R^3R 3 are perpendicular

eq1: ax - 20y + 10z = -1

ew2: ax + ay + 10z = -15.

?x?20y+10z=?1, ?x+?y+10z=?15. Write your answer in the form { a,b} or { a}{a} ?x?20y+10z=?1, ?x+?y+10z=?15. Write your answer in the form { a,b} or { a}{a} ?x?20y+10z=?1, ?x+?y+10z=?15. Write your answer in the form { a,b} or { a}{a} ?x?20y+10z=?1, ?x+?y+10z=?15. Write your answer in the form { a,b} or { a}{a} ?x?20y+10z=?1, ?x+?y+10z=?15. Write your answer in the form { a,b} or { a}{a}

Explanation / Answer

Given planes are:

ax-20y+10z=-1 ----->1

ax+ay+10z=-15 ------>2

The normal vector for plane 1 is given by:
<a,-20,10>
The normal vector for plane 2 is given by:
<a,a,10>

Consider the dot product:

So <a,-20,10>.<a,a,10> = a^2-20a+100

For the planes to be perpendicular

the dot product must be zero

a^2-20a+100=0

(a-10)^2=0

So a=10

If the normal vectors of both planes are perpendicular then the planes themselves are perpendicular

Now consider

?x?20y+10z=?1

?x+?y+10z=?15

The normal vector for plane 1 is given by:
<?,-20,10>
The normal vector for plane 2 is given by:
<?,?,10>

Consider the dot product:

So <?,-20,10>.<?,?,10> = ?^2-20?+100

For the planes to be perpendicular

the dot product must be zero

?^2-20?+100=0

(?-10)^2=0

So ?=10

If the normal vectors of both planes are perpendicular then the planes themselves are perpendicular

final answer is

{a,?} = {10,10}

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