The multiplication of a non-zero complex number by i rotates it through ____________ in the anti-clockwise direction.

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Solution

Let z = x + iy then iz = ix + i²y
i.e iz = ix – y
i.e iz = – y + ix
$\tan θ_{1} for z = x + i y, \tan θ_{1} = \frac{y}{x} \tan θ_{2} for i z = i x - y, \tan θ_{2} = \frac{x}{- y} Since \tan θ_{1} \times \tan θ_{2} = \frac{y}{x} \times \frac{x}{- y} = - 1$
i.e iz is rotated by 90°.

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The multiplication of a non-zero complex number by i rotates it through ____________ in the anti-clockwise direction.

Let z = x + iy then iz = ix + i2y i.e iz = ix – y i.e iz = – y + ix tan θ1 for z=x+iy, tan θ1=yxtan θ2 for iz=ix-y, tan θ2=x-ySince tan θ1×tan θ2=yx×x-y=-1 i.e iz is rotated by 90°.

Let z = x + iy then iz = ix + i²y
i.e iz = ix – y
i.e iz = – y + ix
$\tan θ_{1} for z = x + i y, \tan θ_{1} = \frac{y}{x} \tan θ_{2} for i z = i x - y, \tan θ_{2} = \frac{x}{- y} Since \tan θ_{1} \times \tan θ_{2} = \frac{y}{x} \times \frac{x}{- y} = - 1$
i.e iz is rotated by 90°.