If the complex numbers z1, z2 and z3 are represented on the...

If the complex numbers $z_{1}, z_{2}$ and $z_{3}$ are represented on the Argand diagram by the points $\mathrm{P}_{1}, \mathrm{P}_{2}$ and $\mathrm{P}_{3}$, respectively and $$ \overrightarrow{\mathrm{OP}}_{2}=2 \mathrm{j} \overrightarrow{\mathrm{OP}}_{1} \quad \text { and } \quad \overrightarrow{\mathrm{OP}}_{3}=\frac{2}{5} \mathrm{j} \overrightarrow{\mathrm{P}_{2} \mathrm{P}_{1}} $$ prove that $\mathrm{P}_{3}$ is the foot of the perpendicular from $\mathrm{O}$ onto the line $\mathrm{P}_{1} \mathrm{P}_{2}$

If the complex numbers $z_{1}, z_{2}$ and $z_{3}$ are represented on the Argand diagram by the points $\mathrm{P}_{1}, \mathrm{P}_{2}$ and $\mathrm{P}_{3}$, respectively and
$$
\overrightarrow{\mathrm{OP}}_{2}=2 \mathrm{j} \overrightarrow{\mathrm{OP}}_{1} \quad \text { and } \quad \overrightarrow{\mathrm{OP}}_{3}=\frac{2}{5} \mathrm{j} \overrightarrow{\mathrm{P}_{2} \mathrm{P}_{1}}
$$
prove that $\mathrm{P}_{3}$ is the foot of the perpendicular from $\mathrm{O}$ onto the line $\mathrm{P}_{1} \mathrm{P}_{2}$

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Complex Numbers in the Argand Diagram

This concept involves representing complex numbers as points in a two-dimensional plane, where the horizontal axis corresponds to the real part and the vertical axis to the imaginary part. It allows complex number operations to be interpreted geometrically, providing a visual perspective on translations, rotations, and scaling in the plane.

Complex Multiplication as Rotation

Multiplying a complex number by the imaginary unit (or any unit complex number) corresponds to a rotation in the plane. Specifically, multiplying by the imaginary unit rotates a vector by 90° in an anticlockwise direction. This property is essential in linking algebraic transformations to geometric rotations, particularly when analyzing perpendicularity in the context of complex numbers.

Projection and the Foot of a Perpendicular

The foot of the perpendicular from a point to a line is the unique point on the line at which the line drawn from the external point is perpendicular to the line. This concept is fundamental in geometry and vector analysis, and it is used to determine the shortest distance from a point to a line, often involving the application of dot product properties or rotational transformations in the complex plane.

Transcript

Please give Ace some feedback