how to calculate if two vectors are 'parallel'?

Question

We know we can check if two vectors are 'orthogonal' by doing an inner product.

$a*b=0$

tells us that these two vectors are orthogonal

here comes the question:

if there a way to compute if they are 'parallel'? i.e., they are pointing at the same direction.

Answer 1

Two vectors are parallel iff the absolute value of their dot product equals the product of their lengths.

Iff their dot product equals the product of their lengths, then they “point in the same direction”.

Answer 2

Note that two vectors $\vec v_1,\vec v_2\neq \vec 0$ are parallel $$\iff \vec v_1=k\cdot \vec v_2$$

for some $k\in \mathbb{R}$ and this condition is easy to check component by component.

For vectors in $\mathbb{R^2}$ or $\mathbb{R^3}$ we could check the condition by cross product.

More in general, the matrix formed by two parallel vectors has $rank=1$.

Answer 3

Two parallel vectors $a$ and $b$ will have a null cross product: $$a\times b=0\iff a\parallel b.$$

Answer 4

Two vectors are parallel iff their Gram-determinant, that is the area of the parallelogram spanned by the vectors, vanishes.

Answer 5

Two vectors are parallel iff the dimension of their span is less than $2$.

Answer 6

You can check the followings:

1) Find their slope if you have their coordinates. The slope for a vector $\vec v$ is $λ=\frac{y_v}{x_v}$. If the slope of $\vec a$ and $\vec b$ are equal, then they are parallel.

2) Find the if $\vec a= k\vec b$ where $k \in \mathbb{R}$. If there is a value that satisfies the above equation, then they are parallel.