# Intuition (geometric or otherwise) for: Length of vector product of perpendicular vectors is area of rectangle formed by those vectors

If $$\mathbf a \perp \mathbf b$$, then $$|\mathbf a \times \mathbf b|=|\mathbf a||\mathbf b|$$.

The result can be interpreted as saying that the length of the vector product equals the area of the rectangle that can be formed by $$\mathbf a$$ and $$\mathbf b$$.

Is there any intuition (geometric or otherwise) for this?

Definitions used:

Let $$\mathbf{a}=(a_1,a_2,a_3)$$ and $$\mathbf{b}=(b_1,b_2,b_3)$$. Then

• $$\mathbf{a}\times\mathbf{b}=(a_2b_3-a_3b_2,a_3b_1-a_1b_3,a_1b_2-a_2b_1)$$,
• $$\mathbf{a}\cdot\mathbf{b}=a_1b_1+a_2b_2+a_3b_3$$,
• $$|\mathbf{a}|=\sqrt{a_1^2 +a_2^2 +a_3^2}$$,
• $$\mathbf a \perp \mathbf b$$ if $$\mathbf{a}\cdot\mathbf{b}=0$$.
• Have you tried watching 3blue1brown’s Linear Algebra series on YouTube? Jul 12, 2023 at 6:08
• @bananapeel22: No. If there is a specific segment of a specific video that answers this question, please direct me to that.
– user986614
Jul 12, 2023 at 6:16

It is not much about Intuition. It is the Direct Consequence of the Definition.

Consider this Definition :
$$A \times B = ||A|| \cdot ||B|| \cdot \sin (\theta) \hat{n}$$
where $$\hat{n}$$ is the Unit vector Perpendicular to $$A$$ & $$B$$.

When $$\theta = \pi/2$$ ( vectors are Perpendicular , making a rectangle ) , then $$\sin (\theta) = 1$$.

[[ With that Definition , we can get the vector Components when $$A=(a_1,a_2,a_3)$$ & $$B=(b_1,b_2,b_3)$$ , which will match what you have given. ]]

Now , Parallelogram with Sides $$||A||$$ & $$||B||$$ with angle $$\theta$$ will have that Same Area.

The Height of the triangle (Purple line) is $$||B|| \cdot \sin (\theta)$$
Area of 1 triangle (Blue) is $$||A|| \cdot ||B|| \cdot \sin (\theta) / 2$$
Area of Parallelogram (made of the 2 triangles) is $$2 ||A|| \cdot ||B|| \cdot \sin (\theta) / 2$$

That matches the vector Product Definition.

It is not much about Intuition. It is the Direct Consequence of the Definition.

Definition used here is Common.
Check Page 3 : https://www.mathcentre.ac.uk/resources/uploaded/mc-ty-vectorprod-2009-1.pdf :

Page 7 then gives the "Conversion" from that Definition to this formula :

More Details here : https://en.wikipedia.org/wiki/Cross_product

• That is not the definition I gave.
– user986614
Jul 12, 2023 at 6:28
• I have stated [[ in the Part within Square Brackets like this ]] that "my" Definition will give "your" Definition when we write out the Components ( though that Calculation is simple manipulation or terms ) :: My Definition is not actually mine , it is listed here : en.wikipedia.org/wiki/Cross_product#Definition : In Case you want to use that Definition to Derive your "formula" , you can refer to this Section : en.wikipedia.org/wiki/Cross_product#Computing : It uses the rule of Sarrus to get the Components.
– Prem
Jul 12, 2023 at 8:06
• Added a new Section to my Answer , @user24096 , It might make things a little easier !!
– Prem
Jul 12, 2023 at 8:24