If vector a = i + j + k and vector b = j - k, find a vector c such that vector a x vector c = vector b and vector a . vector c = 3

Question

If \(\vec a =\hat i + \hat j + \hat k\) and \(\vec b = \hat j - \hat k\), find a vector  \(\vec c\) such that \(\vec a \times \vec c = \vec b\) and \(\vec a . \vec c = 3\)

Answer 1

Comparing Left Hand Side and Right Hand Side, we get

and other dot multiplication is zero. We get,

x + y + z = 3 …(iv)

Now, add equations (ii) and (iii), we get

(x – z) + (x – y) = 1 + 1

⇒ x + x – y – z = 2

Put this value of y in equation (i), we get

Equation (i) ⇒ z – y = 0

⇒ 2x – y – z = 2 …(v)

Add equations (iv) and (v), we get

(x + y + z) + (2x – y – z) = 3 + 2

⇒ x + 2x + y – y + z – z = 5

⇒ 3x = 5