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Learn Vectors

Quantities with both size and direction, from forces to velocities.

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A vector is a quantity with both a size and a direction, like a velocity of 30 miles per hour heading north or a force pushing up a ramp. You can draw one as an arrow or write it as components, such as (3, 4), meaning 3 units across and 4 up. Vector math covers adding arrows tip to tail, scaling them, finding their length, and two kinds of products: the dot product, which measures how aligned two vectors are, and the cross product, which gives a perpendicular vector in three dimensions. Learnr builds a vectors course at your level and teaches each idea in short interactive lessons that connect the algebra to pictures.

How to learn Vectors, step by step

Separate scalars from vectors. Speed is a scalar; velocity is a vector. Sort everyday quantities like mass, displacement, temperature and force into the two groups until the difference is instinctive.
Switch between arrows and components. Find the magnitude of (3, 4) with the Pythagorean theorem to get 5, and its direction with inverse tangent. Then convert a vector given as a length and angle back to components using cosine and sine.
Add vectors tip to tail. Draw two displacements end to end and measure the result, then add components to confirm. Walking 3 blocks east and 4 north puts you 5 blocks from where you started.
Learn the dot product two ways. Compute (1, 2, 3) dotted with (4, -5, 6) as 4 minus 10 plus 18, which is 12, and connect it to the formula with magnitudes and the cosine of the angle. A dot product of zero means perpendicular.
Add the cross product in three dimensions. Use the determinant pattern with i, j and k, and the right-hand rule to check direction. The cross product's length equals the area of the parallelogram the two vectors span.
Apply vectors to physics problems daily. Resolve forces on an inclined plane and combine a boat's velocity with a river current. Short daily Learnr lessons, backed by a streak, keep components and products fresh between problem sets.

What you'll learn

Represent vectors as arrows and in component form
Add, subtract and scale vectors graphically and algebraically
Find magnitudes, directions and unit vectors
Compute dot products and angles between vectors
Use cross products in three dimensions for perpendicular vectors and areas

What's inside your Vectors course

A sample path Learnr builds for Vectors. Your real course adapts to your goal and level.

1. Scalars versus vectors and arrow diagrams
2. Components, magnitude and direction
3. Vector addition, subtraction and scalar multiples
4. Unit vectors and the i, j, k notation
5. The dot product and projections
6. The cross product and three-dimensional applications

How long does it take to learn Vectors?

Two-dimensional vectors and components take about 2 to 3 weeks at 3 to 4 hours a week. Adding three dimensions, dot and cross products and physics applications brings the total to roughly 5 to 8 weeks.

What you need before you start

Algebra, the Pythagorean theorem and basic right-triangle trigonometry (sine, cosine and tangent). Coordinate geometry helps.

Common mistakes to avoid

Adding magnitudes instead of components. Two 5-unit vectors at right angles add to about 7.07 units, not 10.
Using the wrong quadrant after inverse tangent, so a vector pointing down and left is reported as pointing up and right.
Expecting the cross product to commute. Swapping the order flips the result to point the opposite way.

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Vectors questions, answered

What is the difference between the dot product and the cross product?

The dot product of two vectors is a number that measures how much they point the same way. The cross product, defined in three dimensions, is a new vector perpendicular to both.

Where are vectors used?

Physics uses them for forces, velocities and fields; computer graphics for positions, lighting and camera directions; and machine learning represents data points and word meanings as vectors.