Vector Calculator

A vector carries both magnitude and direction. In component form $\vec{v} = (v_1, v_2)$ or $(v_1, v_2, v_3)$.

Magnitude (length) is the distance formula applied to the components:

$\|\vec{v}\| = \sqrt{v_1^2 + v_2^2} \qquad \|\vec{v}\| = \sqrt{v_1^2 + v_2^2 + v_3^2}$

Direction angle in 2D, measured counter-clockwise from the positive $x$-axis:

$\theta = \arctan\!\left(\frac{v_2}{v_1}\right), \text{ then add } 180^\circ \text{ if } v_1 < 0$

Back to components from magnitude $r$ and direction $\theta$:

$v_1 = r\cos\theta, \qquad v_2 = r\sin\theta$

Unit vector — same direction, length 1:

$\hat{v} = \frac{\vec{v}}{\|\vec{v}\|}$

This is defined only for $\vec{v} \ne \vec{0}$. Two vectors are equal when their components match; position on the page is irrelevant.

Addition and the resultant

Add component by component:

$\vec{u} + \vec{v} = (u_1 + v_1,\; u_2 + v_2)$

The sum is the resultant — the single vector with the same effect as applying both. Scalar multiplication $k\vec{v} = (kv_1, kv_2)$ scales the length by $|k|$ and reverses direction when $k < 0$. Subtraction is $\vec{u} - \vec{v} = \vec{u} + (-\vec{v})$.

A warning that costs marks: $\|\vec{u} + \vec{v}\| \ne \|\vec{u}\| + \|\vec{v}\|$ unless the vectors point the same way. Always add components first, then take the magnitude.

Dot product — output is a scalar

$\vec{u}\cdot\vec{v} = u_1v_1 + u_2v_2 + u_3v_3 = \|\vec{u}\|\|\vec{v}\|\cos\theta$

Zero means perpendicular. Used for angles, projections, and work.

Cross product — output is a vector, 3D only

$\vec{u}\times\vec{v} = (u_2v_3 - u_3v_2,\; u_3v_1 - u_1v_3,\; u_1v_2 - u_2v_1)$

The result is perpendicular to both inputs, which is how you get a normal vector to a plane. Its length $\|\vec{u}\|\|\vec{v}\|\sin\theta$ equals the area of the parallelogram they span, and order matters: $\vec{v}\times\vec{u} = -(\vec{u}\times\vec{v})$.

• Adding magnitudes instead of components. Two forces of 5 at right angles give a resultant of $\sqrt{50} \approx 7.07$, not 10.
• Ignoring the quadrant in $\arctan$. A calculator returns $(-90^\circ, 90^\circ)$ only, so $(-3, -4)$ and $(3, 4)$ produce the same raw output. Check the signs of the components and add $180^\circ$ when $v_1 < 0$.
• Confusing the two products. The dot product returns a number; the cross product returns a vector and exists only in 3D.
• Reversing the cross product. Each component uses the other two indices in cyclic order, and the middle term $u_3v_1 - u_1v_3$ is the one most often written backwards.
• Forgetting to normalise. A "unit normal" must be divided by its own magnitude — a raw cross product is rarely of length 1.
• Mixing 2D and 3D. The cross product is undefined for two 2D vectors; embed them as $(u_1, u_2, 0)$ first if you need it.