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

Given:

Let

Because is perpendicular to both and .

Then

d1 + 4d+ 2d3 = 0 .......(1)

And

3d1 - 2d2 + 7d3 = 0 ......(2)

And (given)

2d1 - d2 + 4d3 = 15.......(3)

So we have the equations to solve as,
d1 + 4d+ 2d3 = 0
3d1 - 2d2 + 7d3 = 0
2d1 - d2 + 4d3 = 15

From equation (1),
d1 = - 4d2 - 2d3 ........(4)

Putting this value in equation 2 we get,

-12d2 - 6d3 - 2d2 + 7d3 = 0

Therefore,
14d2 = d3 .......(5)

Now putting the value of d3 in equation 4 we get,
d1 = -4d2 - 28d2
d1 = -32d2........(6)

Putting (5) and (6) in equation (3) we get,

-64d2 - d2 + 56d2 = 15
-9d2 = 15
d= -5/3

Now we can find other values as well,

Hence

Hence, the required vector is

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