In Figure 3.5, let r ′ be another vector from O to the line of F. Show that r ′ × F = r × F . Hint: r − r ′ is a vector along the line of F and so is a scalar multiple of F . (The scalar has physical units of distance divided by force, but this fact is irrelevant for the vector proof.) Show also that moving the tail of r along n does not change n ⋅ r × F . Hint: The triple scalar product is not changed by interchanging the dot and the cross.
In Figure 3.5, let r ′ be another vector from O to the line of F. Show that r ′ × F = r × F . Hint: r − r ′ is a vector along the line of F and so is a scalar multiple of F . (The scalar has physical units of distance divided by force, but this fact is irrelevant for the vector proof.) Show also that moving the tail of r along n does not change n ⋅ r × F . Hint: The triple scalar product is not changed by interchanging the dot and the cross.
In Figure 3.5, let
r
′
be another vector from
O
to the line of F. Show that
r
′
×
F
=
r
×
F
.
Hint:
r
−
r
′
is a vector along the line of
F
and so is a scalar multiple of
F
.
(The scalar has physical units of distance divided by force, but this fact is irrelevant for the vector proof.) Show also that moving the tail of
r
along
n
does not change
n
⋅
r
×
F
.
Hint: The triple scalar product is not changed by interchanging the dot and the cross.
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
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