Let , and . Find a vector which is perpendicular to both and , and .

To Find: Find a vector d which is perpendicular to both a and b.

Explanation:

Let us Assume

If a and d are perpendicular then a.d=0

Then,

X+2y+2z=0 …(i)

If b and d are perpendicular then b.d=0

Then,

Where

3x-2y+7z=0 …(ii)

If c and d are perpendicular then c.d=15

Then,

Where

2x-y+4z=15 …(iii)

Now, We have three equations,

X+2y+2z=0 …(i)

3x-2y+7z=0 …(ii)

2x-y+4z=15 …(iii)

We can solve it by Elimination Method

Multiply by 2 in eq (i) and then subtract by (iii)

2x+4y+4z=0

2x-y+4z=15

On subtracting we get

5y=-15

y=-3

Now, put the value of y in equation (i) and (ii)

X+2(-3)+2z=0

X+2z=6 …(iv)

And,

3x-2(-3)+7z=0

2x+7z=-6 …(v)

Now, Multiply by 2 in equation (iv) and subtract by (v), then

2x+4z=12

2x+7z=-6

On subtracting we get

-3z=18

Z=6

Now, Put the value of z in equation (iv), we get

x+2(6)=6

x=6-12

x=-6

Therefore,

Where x=-6 y=-3 and z=6

Hence, This is the required vector.