what is cross product n dot product? please explain well
Answers
Answer:
Cross Product. A vector has magnitude (how long it is) and direction: Two vectors can be multiplied using the "Cross Product" (also see Dot Product) The Cross Product a × b of two vectors is another vector that is at right angles to both: And it all happens in 3 dimensions!
Answer:This completed grid is the outer product, which can be separated into the:
Dot product, the interactions between similar dimensions (x*x, y*y, z*z)
Cross product, the interactions between different dimensions (x*y,y*z, z*x, etc.)
The dot product (
→
a
⋅
→
b
) measures similarity because it only accumulates interactions in matching dimensions. It’s a simple calculation with 3 components.
The cross product (written
→
a
×
→
b
) has to measure a half-dozen “cross interactions”. The calculation looks complex but the concept is simple: accumulate 6 individual differences for the total difference.
Instead of thinking “When do I need the cross product?” think “When do I need interactions between different dimensions?”.
Area, for example, is formed by vectors pointing in different directions (the more orthogonal, the better). Indeed, the cross product measures the area spanned by two 3d vectors (source):
(The “cross product” assumes 3d vectors, but the concept extends to higher dimensions.)
Did the key intuition click? Let’s hop into the details.
Defining The Cross Product
The dot product represents the similarity between vectors as a single number:
\displaystyle{\text{dot product} = (a_x, a_y, a_z) \cdot (b_x, b_y, b_z) = a_x b_x + a_y b_y + a_z b_z = \|\vec{a}\| \|\vec{b}\| \cos(\theta)}
For example, we can say that North and East are 0% similar since
(
0
,
1
)
⋅
(
1
,
0
)
=
0
. Or that North and Northeast are 70% similar (
cos
(
45
)
=
.707
, remember that trig functions are percentages.) The similarity shows the amount of one vector that “shows up” in the other.
Should the cross product, the difference between vectors, be a single number too?
Let’s try. Sine is the percentage difference, so we could write:
\displaystyle{\text{cross product candidate} = \text{amount of difference} = \|\vec{a}\| \|\vec{b}\| \sin(\theta)}
Unfortunately, we’re missing some details. Let’s say we’re looking down the x-axis: both y and z point 100% away from us. A number like “100%” tells us there’s a big difference, but we don’t know what it is! We need extra information to tell us “the difference between
→
x
and
→
y
is this” and “the difference between
→
x
and
→
z
is that“.