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Vector Magnitude and Direction Calculator
Enter a vector's components, or the two points it connects, and mathcheck returns its magnitude as an exact simplified radical and a decimal, its direction angle measured from the positive x-axis (or the three direction angles in 3D), the unit vector in the same direction, and the i, j, k form.
The 2D panel draws the arrow with its components.
Horizontal component of Vector.
Vertical component of Vector.
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Result
Magnitude |v|
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- Direction
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- Unit vector
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- Components
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- i, j, k form
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Magnitude and direction of a vector
A A quantity with both a size and a direction, often drawn as an arrow. has a length (The length or size of a vector, ignoring which way it points.) and a direction. The magnitude comes from the Pythagorean theorem applied to the components. In 2D the direction is the angle θ measured counterclockwise from the positive x-axis; in 3D it is described by three direction angles α, β, γ made with the x-, y-, and z-axes.
Finding the direction angle by quadrant
| Components | Quadrant | Direction angle |
|---|---|---|
| x > 0, y > 0 | I | θ = arctan(y/x) |
| x < 0, y > 0 | II | θ = 180° - arctan(|y/x|) |
| x < 0, y < 0 | III | θ = 180° + arctan(|y/x|) |
| x > 0, y < 0 | IV | θ = 360° - arctan(|y/x|) |
| x = 0 | on the y-axis | 90° if y > 0, 270° if y < 0 |
v = ⟨3, 4⟩
|v| = sqrt(9 + 16) = sqrt(25) = 5. θ = arctan(4/3) ≈ 53.13°. Unit vector: ⟨3/5, 4/5⟩. In i, j form: 3i + 4j.
How to use this calculator
- Choose 2D or 3D.
- Enter the components, or switch to From point P to point Q and enter the two points.
- Read |v| as an exact A root expression written with the √ symbol, such as a square root. with its decimal, and the direction angle (or the three direction angles in 3D).
- Use the unit vector when you need the direction without the length.
- Open Show the work for the substituted formula.
How to read the answer
The magnitude is the length of the arrow, always zero or positive, so ⟨-3, -4⟩ has the same magnitude 5 as ⟨3, 4⟩. The direction angle tells you where the arrow points: 0° is straight right, 90° straight up, 180° left, 270° down. The unit vector has magnitude 1 and the same direction; multiplying it by the magnitude gives the original vector back.
Common mistakes and edge cases
- Using arctan(y/x) without checking the quadrant. ⟨-3, -4⟩ points into Quadrant III at 233.13°, not 53.13°.
- Adding the components instead of squaring them: |⟨3, 4⟩| is sqrt(9 + 16), not 3 + 4.
- Reporting a negative magnitude. Length is never negative; the sign lives in the components.
- Subtracting the points in the wrong order: the vector from P to Q is Q - P.
- The zero vector has magnitude 0 and no direction angle; the calculator flags it.
Worked examples
3-4-5 vector
v = ⟨3, 4⟩
Magnitude |v|
5
Radical magnitude, Quadrant II
v = ⟨-2, 3⟩
Magnitude |v|
√13 (3.6056)
Vector between two points
From P(1, 2) to Q(4, 6)
Magnitude |v|
5
3D vector
v = ⟨1, 2, 2⟩
Magnitude |v|
3
Simplified radical
v = ⟨4, 4⟩
Magnitude |v|
4√2 (5.6569)
Decimal components
v = ⟨1.5, 2⟩
Magnitude |v|
2.5
Pointing straight down
v = ⟨0, -7⟩
Magnitude |v|
7
Zero vector
Magnitude |v|
—
Frequently asked questions
How do I find the magnitude of a vector?+
Square each component, add them, and take the square root. For ⟨3, 4⟩ the magnitude is sqrt(3^2 + 4^2) = sqrt(25) = 5. In 3D include the z-component too.
How do I find the direction of a vector?+
In 2D use θ = atan2(y, x), which gives the angle from the positive x-axis with the correct quadrant. For ⟨-2, 3⟩ that is about 123.69°, not the -56.31° a plain arctan would give.
What is the direction of a 3D vector?+
A single angle is not enough in 3D. Instead we use direction cosines: cos α = x/|v|, cos β = y/|v|, cos γ = z/|v|, giving the angles the vector makes with each axis.
How do I find a vector from two points?+
Subtract the initial point from the terminal point component by component: v = Q - P. From P(1, 2) to Q(4, 6) the vector is ⟨3, 4⟩.
Can magnitude be negative?+
No. Magnitude is a length, so it is zero or positive. Direction is carried by the signs of the components or by the angle.
What is a unit vector?+
A vector of length 1 in the same direction as v, found by dividing v by its magnitude. For ⟨3, 4⟩ the unit vector is ⟨3/5, 4/5⟩.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026