Vectors Explained: Magnitude, Dot Product and Cross Product

A vector has a size (magnitude) and a direction, written with components like $\langle 3, 4 \rangle$. Its magnitude is found with Pythagoras. The dot product multiplies matching components and adds, and tells you the angle between vectors. The cross product of two 3D vectors gives a vector perpendicular to both.

Magnitude

$|\langle 3, 4 \rangle| = \sqrt{9 + 16} = 5$.

The dot product

Example: $\langle 3, 4 \rangle \cdot \langle 1, 2 \rangle = 3 + 8 = 11$.

The angle between two vectors

For the example: $\cos\theta = \frac{11}{5\sqrt{5}} \approx 0.9839$, so $\theta \approx 10.3^\circ$.

The cross product (3D only)

Example: $\langle 1, 2, 3 \rangle \times \langle 4, 5, 6 \rangle = \langle 12 - 15,\; 12 - 6,\; 5 - 8 \rangle = \langle -3, 6, -3 \rangle$.

The result is perpendicular to both vectors, and its length equals the area of the parallelogram they form.

Adding vectors

Add matching components: $\langle 3, 4 \rangle + \langle 1, 2 \rangle = \langle 4, 6 \rangle$. Geometrically, place the second vector at the tip of the first.

Common mistakes

• Expecting the dot product to be a vector (it is a number).
• Getting the order wrong in the cross product: $\mathbf{b}\times\mathbf{a} = -(\mathbf{a}\times\mathbf{b})$.
• Using degrees and radians inconsistently for the angle.

Practice questions

1. $|\langle 6, 8 \rangle|$
2. $\langle 1, 2, 3 \rangle \cdot \langle 4, -5, 6 \rangle$
3. Are $\langle 3, 1 \rangle$ and $\langle -1, 3 \rangle$ perpendicular?

Answers: 1) 10 2) 12 3) Yes, the dot product is 0

The vector calculator finds all of these at once and draws 2D vectors tip to tail. Distance between points uses the same idea; see the distance calculator and the Pythagorean theorem.

Frequently asked questions

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