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Geometric definition

In Euclidean space, a Euclidean vector is a geometric object that possesses both a magnitude and a direction. A vector can be pictured as an arrow. Its magnitude is its length, and its direction is the direction to which the arrow points. The magnitude of a vector a {\displaystyle \mathbf {a} } is denoted by ‖ a ‖ {\displaystyle \left\|\mathbf {a} \right\|}. The dot product of two Euclidean vectors a {\displaystyle \mathbf {a} } and b {\displaystyle \mathbf {b} } is defined by a ⋅ b = ‖ a ‖ ‖ b ‖ cos θ, {\displaystyle \mathbf {a} \cdot \mathbf {b} =\left\|\mathbf {a} \right\|\left\|\mathbf {b} \right\|\cos \theta,} where θ {\displaystyle \theta } is the angle between a {\displaystyle \mathbf {a} } and b {\displaystyle \mathbf {b} }.

In particular, if the vectors a {\displaystyle \mathbf {a} } and b {\displaystyle \mathbf {b} } are orthogonal (i.e., their angle is π 2 {\displaystyle {\frac {\pi }{2}}} or 90 ∘ {\displaystyle 90^{\circ }}), then cos π 2 = 0 {\displaystyle \cos {\frac {\pi }{2}}=0}, which implies that

a ⋅ b = 0. {\displaystyle \mathbf {a} \cdot \mathbf {b} =0.} At the other extreme, if they are codirectional, then the angle between them is zero with cos 0 = 1 {\displaystyle \cos 0=1} and a ⋅ b = ‖ a ‖ ‖ b ‖ {\displaystyle \mathbf {a} \cdot \mathbf {b} =\left\|\mathbf {a} \right\|\,\left\|\mathbf {b} \right\|} This implies that the dot product of a vector a {\displaystyle \mathbf {a} } with itself is a ⋅ a = ‖ a ‖ 2, {\displaystyle \mathbf {a} \cdot \mathbf {a} =\left\|\mathbf {a} \right\|^{2},} which gives ‖ a ‖ = a ⋅ a, {\displaystyle \left\|\mathbf {a} \right\|={\sqrt {\mathbf {a} \cdot \mathbf {a} }},} the formula for the Euclidean length of the vector.