Parallel Vectors Calculator

Enter the vector components to determine if they are parallel or not by calculating their cross-proportions or cross product.

Vector a
⟨ , ⟩
Vector b
⟨ , ⟩

Quick Examples

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Solved Examples

The following are examples of problems solved by the calculator.

Determine if the vectors ⟨2, 4⟩ and ⟨1, 2⟩ are parallel.

Result

The given vectors ARE parallel.

To confirm this, we calculate the equality of the cross-proportions:

$$ \displaystyle a_x b_y - a_y b_x = (2)(2) - (4)(1) = 0 $$

Since the result is zero, vectors a and b are parallel.

Furthermore, vector a can be obtained by multiplying b by the scalar \( \displaystyle k = 2 \):

$$ \displaystyle \langle 2, 4 \rangle = (2) \langle 1, 2 \rangle $$

Since k > 0, the vectors point in the same direction (the angle between them is 0°).

Graph on the Cartesian plane of two parallel vectors pointing in the same direction.
Graph of the vectors
Determine whether the vectors ⟨3, -6⟩ and ⟨-1, 2⟩ are parallel or not.

Result

The given vectors ARE parallel.

To confirm this, we calculate the equality of the cross-proportions:

$$ \displaystyle a_x b_y - a_y b_x = (3)(2) - (-6)(-1) = 0 $$

Since the result is zero, vectors a and b are parallel.

Furthermore, vector a can be obtained by multiplying b by the scalar \( \displaystyle k = -3 \):

$$ \displaystyle \langle 3, -6 \rangle = (-3) \langle -1, 2 \rangle $$

Since k < 0, the vectors point in opposite directions (the angle between them is 180°).

Graph on the Cartesian plane of two parallel vectors pointing in opposite directions.
Graph of the vectors
Calculate the parallelism between the vectors ⟨5, 2⟩ and ⟨1, 1⟩.

Result

The given vectors ARE NOT parallel.

To confirm this, we calculate the equality of the cross-proportions:

$$ \displaystyle a_x b_y - a_y b_x = (5)(1) - (2)(1) = 3 \neq 0 $$

Since the result is non-zero, vectors a and b are not parallel.

Graph on the Cartesian plane of two non-parallel vectors.
Graph of the vectors
Find out whether the vectors ⟨2, 4, -2⟩ and ⟨1, 2, -1⟩ are parallel or not.

Result

The given vectors ARE parallel.

To confirm this, we calculate the cross product:

$$ \displaystyle \vec{a} \times \vec{b} = \begin{vmatrix} \displaystyle \hat{i} & \displaystyle \hat{j} & \displaystyle \hat{k} \\[0.3em] 2 & 4 & -2 \\[0.3em] 1 & 2 & -1 \end{vmatrix} = \langle 0, 0, 0 \rangle $$

Since the resulting vector is the zero vector, vectors a and b are parallel.

Furthermore, vector a can be obtained by multiplying b by the scalar \( \displaystyle k = 2 \):

$$ \displaystyle \langle 2, 4, -2 \rangle = (2) \langle 1, 2, -1 \rangle $$

Since k > 0, the vectors point in the same direction (the angle between them is 0°).

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Daniel Machado

Professor of Mathematics, graduated from the Faculty of Exact, Chemical and Natural Sciences of the National University of Misiones (UNAM). Developer and creator of RigelUp, dedicated to building tools for mathematical learning.

Last updated: (4 days ago)