What is the cross product of two vectors whose result is the zero vector?

[amy098yay]

Homework Statement

1. Verify (https://ucdsb.elearningontario.ca/content/enforced/4850117-BL_1415Sem2__MAT_MCV4UU-948314_1_ELO/MCV4UPU01/MCV4UPU01A06/images/vec-a.gif?_&d2lSessionVal=Y3hirJUTSYjH76OEZwqHIBATE&ou=4850117 + https://ucdsb.elearningontario.ca/content/enforced/4850117-BL_1415Sem2__MAT_MCV4UU-948314_1_ELO/MCV4UPU01/MCV4UPU01A06/images/vec-b.gif?_&d2lSessionVal=Y3hirJUTSYjH76OEZwqHIBATE&ou=4850117) × (https://ucdsb.elearningontario.ca/content/enforced/4850117-BL_1415Sem2__MAT_MCV4UU-948314_1_ELO/MCV4UPU01/MCV4UPU01A06/images/vec-a.gif?_&d2lSessionVal=Y3hirJUTSYjH76OEZwqHIBATE&ou=4850117 + https://ucdsb.elearningontario.ca/content/enforced/4850117-BL_1415Sem2__MAT_MCV4UU-948314_1_ELO/MCV4UPU01/MCV4UPU01A06/images/vec-b.gif?_&d2lSessionVal=Y3hirJUTSYjH76OEZwqHIBATE&ou=4850117) = 0⃗ . What can be said about two vectors whose cross product is the zero vector

a x b

= i j k
3 -3 1
-12 12 -4

=
i
( (-3) · (-4) - 1 · 12 ) -
j
( 3 · (-4) - 1 · (-12) ) +
k
( 3 · 12 - (-3) · (-12) ) =
=
i
( 12 - 12 ) -
j
( (-12) - (-12) ) +
k
( 36 - 36 ) =
=
(0 ; 0 ; 0)

2. Homework Equations

have i correctly verified with an example that two vectors whose cross product is the zero vector?

 

[jasonleroy]

You have created two vectors with zero cross product. Do you understand what can be said about two vectors whose cross product is zero?

 

[amy098yay]

You have created two vectors with zero cross product. Do you understand what can be said about two vectors whose cross product is zero?

They are parallel, since the cross product involves the cosine function. cos(90) = 0. The cross product is a sine function, if it is zero, the angle is zero or 180, so they are in the same or opposite directions. ?

 

[jasonleroy]

That's it, but the cross product is AB*sin(x). I'm not following what you meant by the line below, but you got the main point.

...the cross product involves the cosine function. cos(90) = 0

 

[Mark44]

That's it, but the cross product is AB*sin(x).

No, that's not the cross product. What you're probably thinking of is |A| |B| |sin(θ|), which gives the magnitude of A X B.

 

[Mark44]

They are parallel, since the cross product involves the cosine function. cos(90) = 0.

As already pointed out by jasonleroy, the cross product does NOT involve the cosine function. The cross product also does not involve the sine function, although the magnitude of the cross product does.

The cross product is a sine function, if it is zero, the angle is zero or 180, so they are in the same or opposite directions. ?

 

[amy098yay]

As already pointed out by jasonleroy, the cross product does NOT involve the cosine function. The cross product also does not involve the sine function, although the magnitude of the cross product does.

then what would the cross product be ?

 

[Mark44]

then what would the cross product be ?

Just what you already did with the "pseudodeterminant."

$$\begin{vmatrix} i & j & k \\ x_1 & x_2 & x_3 \\ y_1 & y_2 & y_3\end{vmatrix}$$