2D Vectors

Arrows with magnitude and direction — add them, dot them, understand them

A vector has both a magnitude (length) and a direction. On a 2D graph, vectors are arrows from one point to another. The vector \vec{v} = (3, 4) means "go 3 right and 4 up."

You can add vectors tip-to-tail. The dot product tells you how much two vectors point in the same direction — zero when perpendicular.

In this lesson you'll see vectors as labeled segments, add them graphically, compute magnitudes, and explore the dot product.

Related Topics

Pythagorean TheoremPoint to Line DistanceIntro to Trigonometry

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What is a vector?
A vector has both magnitude and direction. Written as (x, y), it represents a displacement.
How do you find the magnitude?
|\vec{v}| = \sqrt{x^2 + y^2}. For (3, 4): √(9+16) = 5.
How does vector addition work?
Add components: (a_1+b_1, a_2+b_2). Graphically, place tail of b at tip of a.
What is the dot product?
\vec{a} \cdot \vec{b} = a_1 b_1 + a_2 b_2 = |a||b|\cos\theta. Zero means perpendicular.
What can it graph?
It can plot explicit, implicit, and parametric functions, add points and geometry, and animate sliders on the same graph.
Can I use voice or a photo?
Yes. You can talk to the tutor, upload a worksheet or handwritten problem, and let the graph update from that input.
Will it explain the steps?
Yes. The AI explains what it is drawing and why, so you see the answer on the graph instead of getting only a final number.