Why Is Vector Notation Essential in Cross Product Calculations?

[moatasim23]

Why do we use the coordinates of r in terms of x,y,z?Why don't we express coordinates of A in x,y,z?

 

[Simon Bridge]

A is expressed in terms of x y and z.

 

[CompuChip]

It's a matter of notation: we're just giving names to the three components of both vectors.
You can replace them by $r_1, r_2, r_3$, if that makes you feel any better. In general, if v is a vector, it is customary to denote its components by v₁, v₂, v₃. However, if r is the position vector, then (x, y, z) is also quite common.

Also note that though the fact that one is named r hints that it comes from a physical application in which a position vector is involved, the mathematical identity actually holds for any two vectors u, v.

 

[moatasim23]

No its not..here at least

 

[moatasim23]

It's a matter of notation: we're just giving names to the three components of both vectors.
You can replace them by $r_1, r_2, r_3$, if that makes you feel any better. In general, if v is a vector, it is customary to denote its components by v₁, v₂, v₃. However, if r is the position vector, then (x, y, z) is also quite common.

Also note that though the fact that one is named r hints that it comes from a physical application in which a position vector is involved, the mathematical identity actually holds for any two vectors u, v.

If we use r1,r2,r3 then how would the vector operator operator operate on it?Like it didnt in A when we used A1,A2,A3.

 

[SteamKing]

What are you talking about?

 

[Simon Bridge]

No its not..here at least

I'm sorry - the example in your attachment very clearly states that

A=A₁i+A₂j+A₃k

That means that
- the x component of A is A₁,
- the y component of A is A₂,
- the z component of A is A₃.

Therefore: A is resolved in terms of x, y, and z.

What did you think it meant?

 

[CompuChip]

I don't understand your question, I think.

If $\mathbf v = v_1 \mathbf i + v_2 \mathbf j + v_3 \mathbf k$ and $\mathbf u = u_1 \mathbf i + u_2 \mathbf j + u_3 \mathbf k$ then
$$\mathbf u \times \mathbf v = (u_2 v_3 - u_3 v_2) \mathbf i + (u_3 v_1 - u_1 v_3) \mathbf j + (u_1 v_3 - u_3 v_1) \mathbf k$$

That's just how the cross product works. It doesn't matter how you call the components. You could replace $u_1$, $u_2$ and $u_3$ by $x$, $y$ and $z$ or clubs, spades, hearts or bunny, cow, eagle and the definition would still be the same.

Is it the notation of a vector like$\mathbf v = v_1 \mathbf i + v_2 \mathbf j + v_3 \mathbf k$ instead of $\mathbf v = (v_1, v_2, v_3)$ that confuses you?