Vector Algebra

To find the magnitude of vector

^

d

, which is normal to the plane formed by vectors a and b, and also satisfies

^

d

⋅

^

c

=2, we perform the following calculations:
Firstly, calculate a×b, the cross product of vectors a and b :
d=λ(a×b)=λ|

^ i	^ j	^ k
1	1	−2
1	−2	1

|
The calculation proceeds as follows:
d‌=λ(

^

i

(1⋅1−(−2)⋅(−2))−

^

j

(1⋅1−(−2)⋅1)+

^

k

(1⋅(−2)−1⋅1))
‌=λ(

^

i

(1−4)−

^

j

(1+2)+

^

k

(−2−1))
‌=λ(−3

^

i

−3

^

j

−3

^

k

)
‌=−3λ(

^

i

+

^

j

+

^

k

)
Next, use the condition

^

d

⋅

^

c

=2 :
d⋅c=−3λ[2

^

i

+1

^

j

−1

^

k

]
Solves as:
−3λ[2+1−1]‌=2
−3λ×2‌=2
−6λ‌=2
λ‌=−‌

1

3

Substitute λ back to determine d :
d=λ(−3(

^

i

+

^

j

+

^

k

))=

^

i

+

^

j

+

^

k

Finally, the magnitude of

^

d

is:
|d|=√12+12+12=√3