# Let A = [[a, b],[c, d]]be a 2x2 matrix. Let vector v = <a,b> and vector w = <c,d> be the column vectors of A.Compute the cross product of vector v_R3 = <a,b,0> and vector w_R3 =...

Let

A = [[a, b],[c, d]]

be a 2x2 matrix. Let vector v = <a,b> and vector w = <c,d> be the column vectors of A.

Compute the cross product of vector v_R3 = <a,b,0> and vector w_R3 = <c,d,0>

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### 2 Answers

You need to evaluate the cross product of vectors `bar v_(R^3)` and `bar w_(R^3)` , such that:

`bar v_(R^3) X bar w_(R^3) = [(bar i, bar j, bar k),(a, b, 0),(c, d, 0)]`

`bar v_(R^3) X bar w_(R^3) = 0*b*bar i + a*d*bar k + 0*c*bar j - b*c*bar k - 0*d*bar i - 0*a*bar j`

`bar v_(R^3) X bar w_(R^3) = a*d*bar k - b*c*bar k`

Factoring out `bar k` yields:

`bar v_(R^3) X bar w_(R^3) = (a*d - b*c)*bar k`

You should notice that evaluating the determinant of matrix A yields:

`det A = [(a,c),(b,d)] ` => `det A = ad - b*c`

**Hence, evaluating the cross product of vectors `bar v_(R^3)` and `bar w_(R^3)` , yields **`bar v_(R^3) X bar w_(R^3) = det A*bar k.`

let vector (i,j,k) be basis of `R^3`

`vxxw=|[i,j,k],[a,b,0],[c,d,0]|`

`=i(0)+j(o)+k(ad-bc)`

`=k(ad-bc)`

which is required solution.