Verify (a + a) × (a+ b) = 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. Relevant equations
have i correctly verified with an example that two vectors whose cross product is the zero vector?
that looks right....what can be said about them is that they are parallel in space
Yes, you have correctly verified with an example that two vectors whose cross product is the zero vector.
To clarify, in order to prove that the cross product of two vectors is the zero vector, you need to calculate the cross product of the two vectors algebraically. If the resulting vector is the zero vector (0, 0, 0), then you can conclude that the cross product is zero.
In your example, you first calculated the cross product of vectors a and b:
a x b =
(3, -3, 1) x (-12, 12, -4) =
(i, j, k)
((-3) x (-4) - 1 x 12, 3 x (-4) - 1 x (-12), 3 x 12 - (-3) x (-12)) =
(12 - 12, -12 - (-12), 36 - 36) =
(0, 0, 0).
Since the resulting vector is the zero vector (0, 0, 0), you have verified that the cross product of vectors a and b is indeed the zero vector.