where and when should we use dot and cross product and why?

Question

Latika Leekha · Answer

Hello student,
It is extremely important to know the difference between dot and cross products and their usage.
We define the dot product of two vectors
r = ai + bj and s = ci + dj
to be r . s = ac + bd
Cross-product of two vetcors a and b is defined as
(ax, ay, az) X (bx, by, bz) = aybz - azby, azbx - axbz,axby - aybx = ||a||||b|| sin θ n
where n is the unit vector normal to a and b.

The dot product of two vectors is a number and not a vector.
The dot product represents vector similarity with a single number:

(ax, ay, az).(bx, by, bz) = axbx + ayby + azbz = ||a||||b|| cos θ

The dot product just measures the similarity as it only considers the interactions in matching dimensions. Hence, it can be stated to be a simple calculation with 3 components.
The cross-product takes into consideration the cross interactions. So, whenever you need interaction between differnet dimensions, you should use cross-product.
eg: Area is formed by vectors which point in varied directions. Hence, cross-product is used to emasure the area formed by two 3-d vectors.
Some of the instances where cross-product is used are:

Find the direction perpendicular to two given vectors.
Find the signed area spanned by two vectors.
To figure out if two vectors are orthogonal
“Multiply” two vectors when only perpendicular cross-terms make a contribution.

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where and when should we use dot and cross product and why?

where and when should we use dot and cross product and why?

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