Q .4. - Calculus

Question

What does it mean geometrically for two vectors to be orthogonal at a point? What does it mean algebraically? What do we mean when we say that two curves are orthogonal at a point of intersection?

Step-by-Step Solution

Answer

When two curves are orthogonal to each other at a point their tangents at that point make an angle of $90°$ with each other.

Step 1:Given information

 Two vectors to be orthogonal at a point 

Step 2:Explaination

First consider two vectors that are orthogonal at a point.

The geometrical meaning of two orthogonal vectors is that the angle between the vectors is $90°$ at that point when the initial points of the vectors are joined.

This gives that the component of a vector along the other vector is zero.

The algebraic meaning is that the dot product of the two vectors is zero.

This comes from the following fact,

Assume that two vectors $\mathbf{a}$ and $\mathbf{b}$ are orthogonal.

$\mathbf{a}·\mathbf{b}=\Vert \mathbf{a}\Vert \Vert \mathbf{b}\Vert \cos \theta$

$=\Vert \mathbf{a}\Vert |\Vert \mid \mathbf{b}\Vert \cos 90$

$=0$

Now, consider two curves that are orthogonal at a point.

When two curves are orthogonal at a point, it implies that the tangents to the curves at that point are orthogonal to each other.

That is, when two curves are orthogonal to each other at a point their tangents at that point make an angle of $90°$ with each other.