Question

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

Answer 1

The given vectors are a → = i ^ +4 j ^ +2 k ^ , b → =3 i ^ −2 j ^ +7 k ^ and c → =2 i ^ − j ^ +4 k ^ .

Let d → be the vector such that d → =x i ^ +y j ^ +z k ^ .

It is given that d → is perpendicular to both a → and b → .

So, d → ⋅ a → =0and d → ⋅ b → =0.

d → ⋅ a → =0 ( x i ^ +y j ^ +z k ^ )⋅( i ^ +4 j ^ +2 k ^ )=0 x+4y+2z=0 (1)

And,

d → ⋅ b → =0 ( x i ^ +y j ^ +z k ^ )⋅( 3 i ^ −2 j ^ +7 k ^ )=0 3x−2y+7z=0 (2)

Also, it is given that c → ⋅ d → =15.

( 2 i ^ − j ^ +4 k ^ )( x i ^ +y j ^ +z k ^ )=15 2x−y+4z=15 (3)

Solve equation (1), (2) and (3).

x | 4 2 −2 7 | = −y | 1 2 3 7 | = z | 1 4 3 −2 | x 28−( −4 ) = −y 7−6 = z −2−12 x 32 = y −1 = z −14

Write x&y in terms of z.

x 32 =− z 14 x=− 32z 14 x=− 16z 7 (4)

And,

y= z 14 (5)

Substitute the values of x&yin equation (3).

−32 7 z− 1 14 z+ 4 1 z=15 −9 14 z=15 z= −70 3

Substitute the value of z in equation (4) and (5).

x= −16 7 × −70 3 x= 160 3

And,

y= 1 14 × −70 3 y= −5 3

Thus, the required vector d → is,

d → =x i ^ +y j ^ +z k ^ d → = 1 3 ( 160 i ^ −5 j ^ −70 k ^ )