What is the unit vector that is normal to the plane containing ( i +k) and (i+2j+2k) ?
1 Answer
Explanation:
The vector we're looking for is
Using this fact, we can make a system of equations:
vecn * (i+0j+k) = 0
(ai+bj+ck)(i+0j+k)=0
a+c = 0
vecn * (i+2j+2k) = 0
(ai+bj+ck) * (i+2j+2k) = 0
a+2b+2c = 0
Now we have
a+c = a+2b+2c
0 = 2b+c
therefore a+c = 2b+c
a = 2b
a/2 = b
Now we know that
ai + a/2j-ak
Finally, we need to make this a unit vector, meaning we need to divide each coefficient of the vector by its magnitude. The magnitude is:
|vecn|=sqrt(a^2+(a/2)^2+(-a)^2)
|vecn|=sqrt(9/4a^2)
|vecn| = 3/2a
So our unit vector is:
vecn = a/(3/2a)i + (a/2)/(3/2a)j + (-a)/(3/2a)k
vecn = 2/3i + 1/3j -2/3k
Final Answer
