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Vectors and matrices

Vectors

A vector is an arrow with a length and a direction, and equally a list of numbers. This topic adds and scales arrows, asks which points a few arrows can reach by combining them, and measures length and angle with the dot product.

Lessons on this topic

Animation: adding arrows tip to tail, v = (3, 1) and w = (1, 2). W starts at the tip of v. The sum v + w = (4, 3) runs from the origin to the end of w. A faded copy of v is −v = (−3, −1).Animation: adding arrows tip to tail, v = (3, 1) and w = (1, 2). W starts at the tip of v. The sum v + w = (4, 3) runs from the origin to the end of w. A faded copy of v is −v = (−3, −1).

Beginner · 15 minYou are hereRead

Vectors

Read a vector as an arrow and as a list of numbers, add arrows tip to tail, scale them, and see a sound clip as one long vector.

Animation: dots land at c times v plus d times w, v = (2, 1) and w = (1, 2), for weights between −3 and 3. 4800 dots have landed. The combinations fill the whole plane.Animation: dots land at c times v plus d times w, v = (2, 1) and w = (1, 2), for weights between −3 and 3. 4800 dots have landed. The combinations fill the whole plane.

Beginner · 18 minYou are hereRead

Linear combinations and span

Stretch two arrows and add them to see every place they can reach, and spot when a second arrow adds nothing new.

Animation: w turns round and casts a shadow on the line of v = (3, 1). W = (−2.121, −0.707), 180° from v. The shadow is −2.236 long, behind the origin, and v · w = −7.071.Animation: w turns round and casts a shadow on the line of v = (3, 1). W = (−2.121, −0.707), 180° from v. The shadow is −2.236 long, behind the origin, and v · w = −7.071.

Beginner · 22 minYou are hereRead

Length and the dot product

Measure an arrow’s length, see the dot product as a shadow, and use it to find angles and right angles in any number of dimensions.

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Signal processing
Linear algebra
Probability and statistics
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