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The dot product

The dot product of two vectors is a number, not a new vector. You multiply the matching coordinates and add. The same number is the product of the lengths and the cosine of the angle between them. If the dot product is positive, the angle is acute. If it is negative, the angle is obtuse. You use it both to calculate angles and to test whether vectors are perpendicular.

u⃗⋅v⃗=x1x2+y1y2\vec u \cdot \vec v = x_1x_2 + y_1y_2dot product
u⃗⋅v⃗=∣u⃗∣∣v⃗∣cos⁡v\vec u \cdot \vec v = |\vec u||\vec v|\cos vwith the angle

Symbols

u⃗, v⃗\vec u,\ \vec vthe vectors
vvthe angle between them

Example

[2,4]⋅[3,−1]=6−4=2[2, 4] \cdot [3, -1] = 6 - 4 = 2. [1,0]⋅[1,1]=1[1, 0] \cdot [1, 1] = 1, and the cosine of the angle is 1/21/\sqrt{2}, that is 45∘45^\circ.

The dot product is a number. Do not write the answer as a new vector in brackets.
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Part of Mathematics R1: Vectors in the plane.