Calculations:

Apply the distribution property of the cross product, we get

\((i+j)\times(i-j)=i\times i-i\times j+j\times i-j\times j\)

And since the angle between the cross product \(i\times i\) and \(j\times j\) is zero, and since \(\sin(0^\circ)=0\) then from equation the cross product for any two vectors having the same direction is also zero, substituting we get

\(=0-i\times j+j\times i-0\)

\(=-i\times j+j\times i\)

Applying, the commutation property of the cross product, we get

\(=-i\times j-i\times j\)

Applying equation, we get

\(=-2(|a||b|\sin(\theta)n)\)

And since the magnitude of the unit direction i and j is 1, and the angle between both vectors is \(90^\circ\), thus \(\sin\theta=1\) therefor we have

\(=-2(1)(1)(1)n\)

\(i\times j=-2n\)

And, the direction of the unit vector n is orthogonal to \(i\times j\), is determined using the right hand rule, where curling the fingers from the i to j the thumb would be pointing toward the positive direction of k, therefor

\(n=k\)

And, thus the final answer is

\(=-2k\)

Result: -2k

Apply the distribution property of the cross product, we get

\((i+j)\times(i-j)=i\times i-i\times j+j\times i-j\times j\)

And since the angle between the cross product \(i\times i\) and \(j\times j\) is zero, and since \(\sin(0^\circ)=0\) then from equation the cross product for any two vectors having the same direction is also zero, substituting we get

\(=0-i\times j+j\times i-0\)

\(=-i\times j+j\times i\)

Applying, the commutation property of the cross product, we get

\(=-i\times j-i\times j\)

Applying equation, we get

\(=-2(|a||b|\sin(\theta)n)\)

And since the magnitude of the unit direction i and j is 1, and the angle between both vectors is \(90^\circ\), thus \(\sin\theta=1\) therefor we have

\(=-2(1)(1)(1)n\)

\(i\times j=-2n\)

And, the direction of the unit vector n is orthogonal to \(i\times j\), is determined using the right hand rule, where curling the fingers from the i to j the thumb would be pointing toward the positive direction of k, therefor

\(n=k\)

And, thus the final answer is

\(=-2k\)

Result: -2k