Question

Two non zero vectors are parallel if and only if their cross product is 0 . True or False?

Answer 1

Answer: True

Reason

Let u and v represent two nonzero vectors.

The cross product u x v is some third vector perpendicular to both. The absolute value of that, |u x v|, represents the area of a parallelogram formed by vectors u and v.

If vectors u and v are parallel, then they form a straight line and not a parallelogram. In effect, this parallelogram has area 0.

|u x v| = 0 leads to u x v = 0 which shows their cross product is 0.

More technically, the cross product of vector u and vector v is the zero vector in 3space denoted as (0,0,0). It's a vector of length 0.

Notice the zero vector satisfies the conditions that applying the dot product with either u or v gets us 0; hence showing this zero vector is perpendicular to u and v.

Answer 2

The statement is true; two non-zero vectors are parallel or antiparallel if their cross product is the null vector, which occurs when the angle between them is 0° or 180°, making the sine of the angle zero.

Step-by-step explanation:

The statement is true: two non-zero vectors are parallel if and only if their cross product is 0. When vectors are parallel (or antiparallel), the angle (θ) between them is either 0° or 180°. As per the definition of the cross product, also known as the vector product, its magnitude is given by the product of the magnitudes of the two vectors and the sine of the angle between them. Hence, if two vectors are parallel, the sine of the angle is sin(0°) or sin(180°), both of which are equal to 0. This causes the magnitude of the cross-product to be zero. As the direction of the cross product is perpendicular to both original vectors, if the magnitude is zero, the vector itself must be the null vector, representing that the original vectors are indeed parallel or antiparallel.