Question

Find a unit vector perpendicular to A = 2i – 3j + 6k and B = i + j - k.​

Answer 1

Explanation:

We know that cross product of any two vectors yields a vector which is perpendicular to both vectors

∴

for two vectors

→

A

and

→

B

if

→

C

is the vector perpendicular to both.

→

C

=

→

A

×

→

B

=

⎡

⎢

⎣

ˆ

i

ˆ

j

ˆ

k

A

1

A

2

A

3

B

1

B

2

B

3

⎤

⎥

⎦

=

(

A

2

B

3

−

B

2

A

3

)

ˆ

i

−

(

A

1

B

3

−

B

1

A

3

)

ˆ

j

+

(

A

1

B

2

−

B

1

A

2

)

ˆ

k

.

Inserting given vectors we obtain

→

C

=

⎡

⎢

⎣

ˆ

i

ˆ

j

ˆ

k

2

1

1

1

−

1

2

⎤

⎥

⎦

=

(

1

×

2

−

(

−

1

)

×

1

)

ˆ

i

−

(

2

×

2

−

1

×

1

)

ˆ

j

+

(

2

×

(

−

1

)

−

1

×

1

)

ˆ

k

.

=

3

ˆ

i

−

3

ˆ

j

−

3

ˆ

k

.

Now unit vector in the direction of

→

C

is

→

C

∣

∣

∣

→

C

∣

∣

∣

∴

∣

∣

∣

→

C

∣

∣

∣

=

√

3

2

+

(

−

3

)

2

+

(

−

3

)

2

=

√

27

=

3

√

3

Therefore desired unit vector is

1

√

3

(

ˆ

i

−

ˆ

j

−

ˆ

k

)