Verify (Vector a + Vector b) × (Vector a + Vector b) = 0⃗ . What can be said about two vectors whose cross product is the zero vector?
To verify the statement, we need to expand the expression (Vector a + Vector b) × (Vector a + Vector b) using the properties of vector operations.
First, let's expand the cross product using the distributive property:
(Vector a + Vector b) × (Vector a + Vector b)
= (Vector a × Vector a) + (Vector a × Vector b) + (Vector b × Vector a) + (Vector b × Vector b)
Now, let's examine each term individually:
The cross product of a vector with itself, Vector a × Vector a, is always zero. This is because the cross product of two parallel vectors is zero, and a vector is always parallel to itself.
The cross product is anti-commutative, meaning that Vector a × Vector b = - (Vector b × Vector a). Therefore, (Vector a × Vector b) + (Vector b × Vector a) is always zero.
Lastly, the cross product of Vector b with itself, Vector b × Vector b, is again zero for the same reason as Vector a × Vector a.
Therefore, we can simplify the expression to:
(Vector a + Vector b) × (Vector a + Vector b) = 0 + 0 + 0 + 0 = 0⃗
This verifies that the cross product of (Vector a + Vector b) with itself is the zero vector.
Now, let's discuss what can be said about two vectors whose cross product is the zero vector.
If the cross product of two vectors is zero, it means that the two vectors are either parallel or antiparallel to each other.
In other words, if Vector a and Vector b are non-zero vectors and Vector a × Vector b = 0⃗ , it implies that Vector a is parallel to Vector b or Vector a is antiparallel to Vector b.
However, if either Vector a or Vector b is the zero vector (0⃗ ), then the cross product will always be the zero vector regardless of their orientation.